Channel Capacity And Modeling Of Optical Fiber

Communications
Salam Elahmadi, Mandyam D. Srinath, Dinesh Rajan
Department of Electrical Engineering, Southern Methodist University, Dallas, TX 75275
Richard Haberrman
Department of Mathematics, Southern Methodist University, Dallas, TX 75275

Abstract-We use the method of multiple scales borrowed wavelengths are used to simultaneously carry data from
from perturbation theory to derive a new time-domain multiple channels. The basic architecture of these systems
transfer function of the nonlinear fiber-optic wave-division is depicted in figure 1. Light from all channels is amplified
multiplexing (WDM) communications channel. The obtained at several sites along the line. The transmitted data is
channel response, derived from the nonlinear Schrodinger
recovered by demultiplexing and converting the received
equation is shown to be equivalent to the multi-path fading
frequency selective channel encountered in wireless links. In light back into the electrical domain.
the linear regime, the channel response is shown to be The focus of this paper is to answer the following basic
equivalent to a standard intersymbol interference (lSI) question: What is the maximal error-free data rate that can
channel and is used to derive new bounds on the capacity of be expected from a fiber optic medium? While Shannon's
the dispersive optical fiber channel. results address this question in a general sense, with the
exception of the additive white Gaussian noise (AWGN)
model, computing the theoretical information capacity has
I. INTRODUCTION proved to be mathematically challenging for many
Over the past few decades, fiber optic networks have channels. The fiber nonlinearities in WDM networks allow
experienced a remarkable evolution. Their bandwidth and for coupling amongst channels, giving rise to undesired
reach have increased nearly a thousand-fold. Earlier effects such as stimulated Raman scattering (SRS), cross-
networks used multi-mode lasers, multi-mode fibers with phase modulation (XPM), and four-wave mixing (FWM).
losses exceeding 2.5 dBlkm, and direct modulation Any attempt to derive the information capacity of the
techniques. They carried data streams at modest rates (a highly nonlinear WDM channel based on the AWGN
few Mb/s) over modest distances (a few kilometers). assumption will evidently yield inaccurate results. The
Today's optical networks are capable of carrying traffic at right approach is to obtain a model that takes into account
hundreds of Gb/s over a few thousand kilometers. Many the physical properties of the fiber optic WDM channel. In
factors have contributed to this marked increase in [1-4]), solutions to this problem were proposed but effects
performance. Chief among them are deployment of single- such as Raman scattering , optical-to-electrical (OE)
mode fibers with losses below 0.2 dBlkm and use of regeneration, or dispersion were either ignored or treated in
distributed feedback (DFB) lasers, external modulation isolation. In sections II and III, we use the generalized
techniques, dispersion compensating fibers, forward error nonlinear Schrodinger equation (NLSE) governing pulse
correction (FEC) , and photonic amplification. The propagation in an optical fiber to derive new channel
migration from single-channel to multi-channel systems models. We show that the channel response can be
has also contributed greatly to boosting the capacity of modeled as a fading, multipath, frequency-selective
today's optical networks. In wave-division multiplexing channel in the nonlinear
(WDM) , a widely deployed technology, multiple

978-1-4244-3474-9/09/$25.00 ©2009 IEEE

 but it must also include a mapping between the initial pulse
A(O,t) and the pulse A(z,t) after it has propagated some
distance z. Unfortunately, the NLSE equation has no

I. known closed-form solution. To proceed further, the

following assumptions are in order. The XPM term in (1) is
twice as important as the SPM term. Moreover, as the
number of channels in a WDM system increases, the XPM
term will grow even larger and will in fact come to
dominate so that SPM can be neglected. The second term
on the left-hand side of (1) indicates that different
Fig. 1. Basic architecture of a WDM system. wavelengths travel at different speeds. After a certain
distance called pulse walk-off, pulses on different channels
regime and as an lSI channel in the linear regime. In walk away from each other and the crosstalk caused by
sections IV and V, we make use of the rich literature their interaction diminishes. In the work that follows, we
treating the Shannon capacity of such a channel to derive will neglect the group delay term in (1) thereby assuming a
new capacity bounds for the linear dispersive optical worst-case scenario .. With these assumptions, equation (1)
channel. Capacity bounds based on our solution to the simplifies to
nonlinear channel will be presented in a future publication.

8A j +~f1 8 2 Aj =GA (2)

8z 2 2j 8t 2 j ,
II. PROPAGATION MODEL
where
Under the slowly-varying-envelope approximation which
is valid for pulse widths greater than 1ps, and ignoring
higher order dispersion coefficients, pulse propagation in G aj "
= --+ L.(- - + 2Yn 1')1 An 12
gnj (3)
the fh channel of aN-channel WDM system is governed by 2 n*j 2
the following equation
Neglecting SPM has allowed us to convert the nonlinear
partial-differential equation (pde) in (1) into a variable-
coefficient linear pde (2). Equation (2) is still a complex
mathematical entity with no trivial solution, nevertheless its
(1)
solution will be shown to be mathematically tractable.

III. LARGE SPACE-TIME SOLUTION TO THE LINEARIZED

where SCHRODINGER EQUATION
Envelope of the jth optical channel
Equation (2) is a linearized version of the nonlinear
Group delay of the fh channel
Schrodinger equation and will be referred to, from hereon,
Group velocity dispersion (GVD)
t
parameter of the channel
as the linear Schrodinger equation (LSE). Although it is a
linear equation, it admits no trivial solution because the
Loss coefficient of the fh channel damping function G(z,t) defmed in (3) is time-dependent.
Nonlinearity parameter for the fh channel Nevertheless, the method of multiple scales borrowed from
Raman coupling coefficient between perturbation theory [6] can be used to find an asymptotic
channels j and n. solution. Accordingly, we first proceed to solve the
unperturbed problem by neglecting the damping function.
Equation (1) is known in the literature as the generalized That is we start by solving the following linear problem
nonlinear Schrodinger equation [5] (NLSE). The third and (the subscriptj is dropped for simplicity)
fourth terms on the left of side of equation (1) account for
the linear effects: dispersion and fiber loss. The first term
on the right-hand side is responsible for both self-phase (4)
modulation (SPM) through the pulse magnitude in the fh
channel, and cross-phase modulation XPM through the
Using Fourier analysis, the general solution of equation (4)
pulse magnitudes from other channels, while the last term
is of the form
accounts for SRS crosstalk. In order to compute the
information capacity of the channel described by (1), not
only must a closed-form solution of equation (1) be found
 1 jA(O,w)exp -P2W2Z-iwt dw,
A(z,t)=-
21£
00

2
-00
- (i ) (5) (13)

where A(O,w) is the Fourier transfonn of the initial pulse Finally, using (10), (12), and (13) in (7) and ignoring the
A(O,t ) (launched at z = 0), defined as constant phase, rjJ, we get

co

A(O,w) = JA(O,t)exP(iwt)dt. (6)

-co (14)

If the input pulse A(O,t) travels a sufficiently large distance

and is allowed to disperse, equation (5) will simplify, under Equation (14) shows that the initial pulse will decay as the
the stationary phase approximation, to inverse square root of the dispersion-distance product but
its shape will remain unchanged.
2 -
A(z,t) "" 1£ A(O,{Oo)exp(ikz - i{Oot - irjJ) As a simple validity check of equation (14), consider the
k"({Oo)z case of a Gaussian pulse for which A(O,t) is given as
(7)
rjJ = sgn(k" (w o»)!:.
4
A(O,t) = exp(-4) , (15)
2To
where k satisfies the dispersion relation

where To is the pulse width at the lie-intensity point. The

(8) Fourier transfonn of this pulse is given as

k"(wo) = d2~ I
dw
,and {oo is the solution to the following
(16)

())o
equation By substituting (16) in (5) and carrying out the integration,
the exact complex envelope at any distance z is found to be
dk
-z-t=O
d{O , (9)
(17)
where dw /dk is the envelope group velocity. Solving for k
in (8) yields
To obtain the stationary phase solution for a Gaussian input
pulse, we evaluate (16) at {oo given in (13) and substitute
(10)
the result in (14) to get

Hence, the first and second derivatives of k are

(18)

(11)
It can easily be shown that when

(12) (19)

Substituting the result from (11) in (9) and solving for (oo, the exact solution given by (17) will converge to the
expression in (18). Figure 2.1 shows magnitude plots of
we obtain
equations (18) and (17) for different fiber lengths assuming
To = 10 ps and /32 = 17 ps21Km. It is clear from the plots
that when condition (19) is satisfied, as is the case for a
 t2
o.7.-----~---~---~---__,
O(z,t) = - - - . (21)
-- Stationary phase solution 2/32 z
0.6 - Exact solution
Let us denote the first and second partial derivatives of a
0.5 function F with respect to x as Fx and Fxx, respectively.
Then,
..,0.4
.~

~ 0.3
(22)
0.2

0.1
where
~1'=-0-----::--5--------:-0---~:---------"10
//TO

(a)

(23)
1.8

-- Stationary phase solution

1.6
- Exact soluti on

1.4

1.2

..,
.u; 1 Substituting (22) in (2), we get
§
.= 0.8

0.6 iOzR+R z +i/32 ~tt -O/R)- /32 (20t R t + Ott R ) = GR.

2 2
0.4
(24)
0.2 From the look of it, one might think, and rightly so, that
0
-10 -5 10
equation (24) is mathematically more difficult to deal with
//TO than equation (4). Because of the terms 0/ and Rtt, equation
(24) admits no trivial solution. However, the dispersion
relation in (8) implies that

(b)
(25)
Fig. 2. Exact and stationary phase solutions of equation (4) when relation
(19) is (a) satisfied and (b) (19) violated.
Moreover, after a sufficiently large length z, the input pulse
will have dispersed and the term Rtt can be neglected
fiber length z = 20 Km (Figure 2.b), the stationary phase without loss of generality. Hence, equation (24) simplifies
solution coincides nearly with the exact solution given by to
(17). For the smaller length z = 2 Km, (Figure 2.a), the
stationary phase approximation does not match the exact
(26)
solution as condition (19) no longer applies. Having solved
the linear problem (4), we now turn our attention back to
the LSE problem defined by equations (2-3). Let us assume We now call on the method of characteristics to solve for R
that a solution to equation (2) exists and is of the form in (26). Recall that R is a function of both space (z) and
time (t) and so its spatial derivative, for an observer
A(z,t) = R(z,t)eiB(z,t) , (20) moving with the envelope, can be expressed as

where R(z,t) is an unknown complex function and ~z,t) is dR =R R ot (27)

the phase obtained from the stationary phase solution, i.e., dz z+az t·
By analogy with (27), solving equation (26) can be reduced IV. CHANNEL MODEL
to solving the following ordinary differential equation Consider a WDM system as depicted in Fig. 1. The
(ode) combined signal travels over a dispersive medium and is
dR = R(G + /32 (J ) (28) amplified before being regenerated at the receiver. Optical
dz 2 It , amplifiers, like any amplifier, not only amplify signals but
they also corrupt them with additive noise. In the case of an
along the characteristic equation erbium-doped amplifier (EDFA), a quasi-essential device
in today's WDM systems, the added noise is attributed to a
physical process known as amplified spontaneous emission
(29) (ASE). ASE noise is commonly modeled as an additive
white Gaussian (AWG) stochastic process with a variance
Replacing (J/ in (29) with the expression in (23) then
integrating yields the following space-time relation O"~SE = nsphvc (G -l)Bo , (33)

t = bz, (30) where nsp is the spontaneous emission factor, h is Planck's

constant, Vc is the optical frequency, G is the gain, and Bo
where b is constant along the characteristic path described is the optical bandwidth. Let Xk represent the symbols
by (29). The o.d.e. in (27) can now be solved for any transmitted over a WDM optical channel and let h(t,L) be
space-time point satisfying relation (30). Replacing (JtI in the combined transmitter and channel response.
(28) with the expression in (23) then integrating yields Furthermore, let the received signal, y(t), be sampled at
t=nT where T is the symbol period. Then
- t
A(O,-) [" ) (34)
R(z,t) = J;;z exp fG(x,bx)dx , (31)
/32 z 0
where Wk represents the ASE noise generated by the optical
amplifier. In the nonlinear regime, the combined sampled
- t channel response given as
where A(O,-) is the Fourier transform of the initial
/32 z
- nT
pulse, evaluated at OJ = _t_ and b = tlz. Finally, using hn = A(O, -)p(L, nT), (35)
/32 z /32L
expressions (31) and (21) in equation (20), we obtain where

A(z,t) =
A(O,_t) (Z J [ 2)
J;;z exp fG(x,!..x)dx exp - i_t- (32)
(36)

/32 z 0 z 2/32 z

Equation (32) is a large-space solution to the LSE equation. is the nonlinear channel distortion, The term, A(O, nT )
It shows that as the initial pulse propagates along the fiber,
/32 z
both its shape and phase (recall that G is a complex causes lSI and p(L,nT), a random multiplicative noise
function) will be altered. Its magnitude will decay as the coefficient, contributes to channel fading. A nonlinear
WDM channel can thus be modeled as a multi-path, fading,
inverse square root of the dispersion-distance product and
frequency selective channel. This simple yet significant
will be modulated, along a given characteristic path, by the
observation can be exploited to derive new bounds on the
fiber nonlinearities. It also provides a closed-form mapping
between the initial pulse A(O,t) and the output pulse A(z,t) capacity of the nonlinear WDM optical fiber channel. In
at any sufficiently large distance z. It thus constitutes a the linear regime, the combined response h(nT, L) is given
large-space (slightly different paths result from removing by (14) and the channel in (34) reduces to the classical
discrete-time intersymbol interference channel.
the large-space condition dictated by the stationary phase
assumption) time-domain transfer function for the
V. CAPACITY BOUNDS
nonlinear WDM fiber optic channel. This result can be
exploited to formulate the theoretical capacity problem of In this work, we limit our discussion to estimating the
the nonlinear WDM communication system in an capacity of the linear optical channel defmed by (14) and
information theoretic framework.
 3.5 r---~---~--~------,
(34). The capacity of the nonlinear channel defined by (34)
(1 )
and (35) will be presented in a future publication.. 3

2.5
The channel capacity is defined as o
~ 2 (2
a;-
C(Y;X) = max/(Y;X) ,
ptA)
(37) 15 1.5
U

where /(Y;X) is the mutual information between Y and X, 0.5

(4)
defmedas

/(Y;X) = E{lO{ Ef;~:;;)}J}. (38) o~====~--~--~--~

-10 -5 0 5 10
SNR(dB)
Fig. 3. Capacity bounds on linear fiber optic communication. (I) Cawg , (2)
Ca, (3) Cb , (4)CLWAfF'
The maximization in (37) is carried out over the probability
densities of the transmitted signal xl subject to the power VI. CONCLUSION
constraint
The treatment of a fiber optic channel as a time-varying
frequency selective channel has, to our knowledge, never
(39) been proposed. It bridges a long-standing gap between the
physics of fiber optics and the theory of communication
and opens fiber optic communication to powerful
where P is the total available power. techniques already used in wireless systems. Estimating
the capacity of a fiber optics system is a fundamental step
In the linear regime where the classical discrete-time to understanding how its performance may be improved.
intersymbol interference channel model holds, the channel We have used the method of multiple scales to solve the
capacity is given as [6] nonlinear Schrodinger equation. The proposed solution is
valid only after the initial pulse has traveled a sufficiently
large distance and is allowed to disperse. In practice this
(40) condition is usually met (as is the case for single mode
fibers with a loss coefficient of 0.22 dBlKm and a GVD
around 17 pslKm) after the pulse has propagated tens of
where Rh is the discrete Fourier transform of hn and P the kilometers. We then used the linear solution to derive new
launch power. The channel capacity in (40) is achieved bounds on the capacity of linear but dispersive optical fiber
only when the input symbols Xk are i.i.d. Gaussian random channel. The capacity of the nonlinear channel based on
variables. In many communications systems this condition our solution (35) will be presented in a future publication.
may not hold. Practical lower bounds however exist to
evaluate the capacity of these systems [7]. One such bound, REFERENCES
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