1

Energy Conservation in Optical Fibers with
Distributed Brick-Walls Filters

arXiv:1707.05018v2 [] 29 Dec 2017

Javier Garcı́a, Hassan Ghozlan, Member, IEEE, and Gerhard Kramer, Fellow, IEEE
Abstract—A band-pass filtering scheme is proposed to mitigate
spectral broadening and channel coupling in the Nonlinear
Schrödinger (NLS) fiber optic channel. The scheme is modeled
by modifying the NLS Equation to include an attenuation profile
with multiple brick-wall filters centered at different frequencies.
It is shown that this brick-walls profile conserves the total in-band
energy of the launch signal. Furthermore, energy fluctuations
between the filtered channels are characterized, and conditions on
the channel spacings are derived that ensure energy conservation
in each channel. The maximum spectral efficiency of such a
system is derived, and a constructive rule for achieving it using
Sidon sequences is provided.
Index Terms—Optical Kerr effect, Optical coupling, Energy
conservation, Bandlimited communication

This paper is organized as follows. Section II describes
the channel model and the brick-walls filters. Section III
proves total energy conservation, both in the frequency and
time domains. Section IV characterizes the energy fluctuations
between channels and gives a constructive rule for designing
systems with per-channel energy conservation. Section V
concludes the paper and gives directions for future work.
II. S YSTEM M ODEL
Consider the slowly varying component q(z, t) of an electrical field propagating along an optical fiber, where z is distance
and t is time. The field obeys the Nonlinear Schrödinger
equation (NLSE), which is expressed as [3, Eq. (2.3.46)]:
∂
α
β2 ∂ 2
q(z, t)
q(z, t) = − q(z, t) − j
∂z
2
2 ∂t2
2
+ jγ |q(z, t)| q(z, t)

I. I NTRODUCTION
The Nonlinear Schrödinger Equation (NLSE), which governs waveform propagation in fiber optic channels, exhibits
signal-noise mixing and channel coupling that limit the achievable communication rates [1]. In this work, we consider
a modified NLSE with a distributed brick-walls filter that
controls spectral broadening. In practical terms, this channel
is a limiting case of a system that includes band-pass filters at
regular spacings along the fiber, and when the filter spacing
tends to zero. We prove that this band-limited system is
energy-preserving. We further characterize the energy fluctuations between the filtered channels and show that avoiding
Four Wave Mixing (FWM) conserves the per-channel energy.
More precisely, we place the channels using Sidon sequences,
as done in [2], and we derive an asymptotic upper bound on
the spectral efficiency.
The per-channel energy conservation suggests that filters
placed at close spacings, or fibers with high attenuation outside
the bands of interest, might increase communication rates at
high powers. In fact, the lack of interference might allow
pulse amplitude modulation rates to grow with launch power,
avoiding the peak reported in [1]. However, our results do
not prove this because we measure energy by integrating over
all time, and this prevents us from characterizing temporal
broadening due to dispersion. We currently lack the analysis
tools for analyzing energy over finite time intervals.

(1)

where α is the attenuation coefficient, β2 is the dispersion
coefficient, and γ is the nonlinear coefficient. Taking the
Fourier transform of (1), we obtain


∂
γ
α(ω)
β2 2
Q(z, ω) = −
+ j ω Q(z, ω) + j 2
∂z
2
2
4π
Z∞ Z∞
·
Q(z, ω1 )Q∗ (z, ω2 )Q(z, ω − ω1 + ω2 ) dω2 dω1 (2)
−∞ −∞

where ω is the angular frequency, Q∗ denotes the complex
conjugate of Q, and
Q(z, ω) =

Z∞

q(z, t)e−jωt dt.

(3)

−∞

To obtain (2), we used the standard result that a product
of signals in the time domain becomes a convolution in the
frequency domain. The convolution of two signals is
f (t) ∗ g(t) =

Z∞

−∞

f (τ )g(t − τ ) dτ.

(4)

The nonlinear term γ causes variations in the spectral
occupancy of signals. In a system with band-pass filters at
discrete positions along the fiber, every filter removes part of
the energy of the signal. One idea to mitigate the loss is to
make the spacing ∆z between the filters small. We prove in
Section III that, in the limit when ∆z → 0, the energy in
the passband is preserved as the signal propagates along the
fiber. An intuitive explanation is that, in a system with lumped
filters at spacings ∆z, the out-of-band energy produced by the
2
nonlinearity grows in proportion to (∆z) [J]. The energy loss

Date of current version January 1, 2018. J. Garcı́a was supported by the
German Research Foundation under Grant KR 3517/8-1. H. Ghozlan was
supported in part by a USC Annenberg Fellowship and in part by the National
Science Foundation under Grant CCF- 09-05235. G. Kramer was supported by
an Alexander von Humboldt Professorship endowed by the German Federal
Ministry of Education and Research.
J. Garcı́a and G. Kramer are with the Institute for Communications Engineering (LNT), Technical University of Munich, 80333 Munich, Germany.
H. Ghozlan was with the Department of Electrical Engineering, University
of Southern California, Los Angeles, CA 90089, USA. He is now with Intel
Corporation, Hillsboro, OR 97124, USA.
This work has been accepted for publication in IEEE/OSA Journal of Lightwave Technology. Digital Object Identifier: 10.1109/JLT.2017.2785316
Copyright 0733-8724 c 2017 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or
future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or
redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.

2

per unit distance is therefore linear in ∆z [J/m], and goes to
0 as ∆z → 0.
In (2), we have allowed for a frequency-dependent attenuation profile α(ω). One way to model the distributed filtering is
to design α(ω) to be small in the bands of interest, and very
large outside them. In the ideal case, we have
(
α0 , ω ∈ S;
α(ω) =
(5)
∞, otherwise
where S denotes a specified set of angular frequencies. We
call (5) a brick-walls attenuation profile. The corresponding
brick-walls filter for α0 = 0 is
(
1, ω ∈ S;
HS (ω) =
(6)
0, otherwise.
For example, consider a generic WDM system with N channels. For each channel Wn , n ∈ {1, . . . , N }, let ωn be its
center frequency, and Wn its width. The band of interest is
W=
where

N
[

n=1

Wn

(7)



Wn
Wn = ω : |ω − ω n | ≤
2

(8)

and
Wm ∩ Wn = ∅,

if m 6= n.

(9)

Proof. Multiplying (2) by Q∗ (z, ω)/(2π) and integrating over
ω, we obtain
1
2π

Z∞

Q∗ (z, ω)

−∞

1
=
2π

∂
Q(z, ω) dω
∂z


Z∞ 
γ
α(ω)
β2 2
2
−
+ j ω |Q(z, ω)| dω + j IR (z)
2
2
π

−∞

(13)
where, for a set S, we defined
Z Z ∞ Z ∞
1
Q(z, ω1 )Q∗ (z, ω2 )
IS (z) =
4π 2 S −∞ −∞
· Q(z, ω − ω1 + ω2 )Q∗ (z, ω) dω1 dω2 dω
Z ∞
1
[Q(z, ω3 ) ∗ Q(z, ω3 )]
=
4π 2 −∞
∗
· [QS (z, ω3 ) ∗ Q(z, ω3 )] dω3
(14)
where ∗ denotes convolution in ω3 . We used ω3 = ω +ω2 , and
defined QS (z, ω) = HS (ω)Q(z, ω). Taking real parts of (13),
and using ℑ {IR (z)} = 0, we obtain (12).

Theorem 2. Let Q (z, ω) ∈ L2 be the Fourier transform of a
bounded signal propagating according to (2), with attenuation
profile (5) and α0 ≥ 0. Then the signal has no energy outside
the band W for z > 0:
EW (z) = 0,

z>0

(15)

The channels are thus pairwise disjoint frequency intervals.
The brick-walls filter H(ω) , HW (ω) for α0 = 0 has an
impulse response given by


N
X
Wn jωn t
Wn
h(t) =
sinc
e
t
(10)
2π
2π
n=1

where W is the complement of W in R. Furthermore, the
signal energy E(z) evolves in z according to

III. E NERGY CONSERVATION

Substituting (17) in (12), we have
d
E(z) = −α0 EW (z) − α1 EW (z).
(18)
dz
Multiplying (2) by Q∗ (z, ω)/(2π), integrating over ω ∈ W
and taking real part, we obtain
γ
d
(19)
E (z) = −α1 EW (z) − ℑ {IW (z)} .
dz W
π
By assumption, the time-domain signal q(z, t) is bounded.
Let |q(z, t)| ≤ qmax . We have

Z ∞
Z ∞Z ∞
1
∗
|IW (z)| =
[HW (ω)Q (z, ω)]
Q(z, ω1 )
4π 2 −∞ −∞
−∞

· Q∗ (z, ω2 )Q(z, ω − ω1 + ω2 ) dω1 dω2 dω
Z ∞
(a)
2
∗
(z, t) |q(z, t)| q(z, t) dt
(20)
= 2π
qW

where sinc(x) , sin(πx)/(πx).

We prove energy conservation in the frequency domain in
Section III-A, and in the time domain in Section III-B. We
write the energy in a frequency band S as
Z
1
ES (z) ,
|Q(z, ω)|2 dω.
(11)
2π S

We write q(z, t) ∈ L2 (or Q(z, ω) ∈ L2 ) if the signal energy
E(z) = ER (z) exists in the Lebesgue sense and E(z) < ∞.
Let ℜ {x} and ℑ {x} be the real and imaginary parts of x,
respectively.
A. Energy Conservation in the Frequency Domain
We characterize the energy evolution for a bounded α(ω).
Lemma 1. Let Q (z, ω) ∈ L2 be a signal propagating
according to (2), with attenuation profile α(ω) < ∞. The
signal energy E(z) evolves in z according to
1
d
E(z) , −
dz
2π

Z∞

−∞

2

α(ω) |Q(z, ω)| dω.

(12)

E(z) = EW (0)e−α0 z .
Proof. Let α1 > 0 and consider the attenuation profile
(
α0 , ω ∈ W
α(ω) =
α1 , otherwise.

(16)

(17)

−∞

where qW (z, t) is the inverse Fourier transform of QW (z, ω),
and (a) follows from Plancherel’s formula [4, Eq. (2.1.2)]
Z ∞
Z ∞
X(ω)Y ∗ (ω) dω.
(21)
x(t)y ∗ (t) dt =
2π
−∞

−∞

3

Using the standard result |hx, yi| ≤ kxk kyk, we obtain
sZ
Z ∞
∞
(b)
2
|qW (z, t)| dt
|q(z, t)|6 dt
|IW (z)| ≤ 2π
−∞

≤ 2π

sZ

q(z + ∆z, t) = e∆z(D̂+Â+N̂ ) q(z, t)

−∞

∞
−∞

respectively. Consider a small step ∆z. The solution of (28)
is

|qW

(z, t)|2

dt

Z

≈ e∆zD̂ e∆zÂ e∆zN̂ q(z, t)

∞

−∞

4
qmax
|q(z, t)|2

q
p
2
E(0) EW (z).
≤ 2πqmax

dt

where the approximation is valid up to first order in ∆z [3,
p. 48]. Consider the dispersion step:

(22)

The last inequality follows from (18) and α(ω) ≥ 0, so that
E(z) ≤ E(0) for z ≥ 0. Using (22) and −ℑ {x} ≤ |x|, we
upper bound (19) as
q
p
d
2
E(0) EW (z). (23)
EW (z) ≤ −α1 EW (z) + 2|γ|qmax
dz
The solutions to a differential inequality can be bounded by
the solution to the corresponding equality [5, p. 7]. This yields
EW (z) ≤ EW (0)e

−α1 z

p
p

α1
E(0) EW (0) − α1 z 
1 − e− 2 z
e 2
+
α1

2
4
α2
4|γ|2 qmax
E(0)
− 21 z
1
−
e
+
.
(24)
α21
2
4|γ|qmax

Letting α1 → ∞ proves (15), and therefore E(z) = EW (z)
for z > 0. From (24), we have limα1 →∞ α1 EW (z) = 0. Using
this in (18) proves (16).
Theorem 2 shows that if the launch signal has energy inside
W only, then there is no energy loss due to the filters. The
exponential energy loss is due to attenuation only.

q(z + ∆z, t) , F̂ −1 ej∆z

−∞

2

|h(t) ∗ q(z, t)| dt

Z∞

E(z) = EW (0)e

−∞

2

·e

z > 0.

2

q(z, t1 )q ∗ (z, t2 )

−∞ −∞ −∞

∆z [jγ (|q(z,t1 )|2 −|q(z,t2 )|2 )]

(33)

As q(z, t) is band-limited to W for z > 0, we have
Z∞

−∞

2

|q(z, t)| dt.

(34)

The derivative of the energy is
E(z + ∆z) − E(z)
∆z
Z∞ Z∞ Z∞
= lim
q(z, t1 )q ∗ (z, t2 )

= lim

∆z→0

∆z→0
−∞ −∞ −∞

(26)

(27)

e∆z[−α0 +jγ (|q(z,t1 )| −|q(z,t2 )| )] − 1
∆z
· h(t − t1 )h∗ (t − t2 ) dt2 dt1 dt
= − α0 E(z) + I1 + I1∗
(35)

where we used lim∆z→0 eb∆z − 1 /∆z = b and
2

Z∞ Z∞

−∞ −∞

2

q(z, t1 ) |q(z, t1 )| h(t − t1 )

·

Z∞

−∞

(28)
2

∂
where N̂ = jγ |q(z, t)| , Â = − 21 a(t)∗, and D̂ = −j β22 ∂t
2
are the nonlinearity, attenuation and dispersion operators,

2

·



Proof. We use a split-step approximation. We write (25) as
∂
q(z, t) = (D̂ + Â + N̂ )q(z, t)
∂z

Z∞ Z∞ Z∞

· e−∆zα0 h(t − t1 )h∗ (t − t2 ) dt2 dt1 dt.

I1 = jγ
,

−∞

|qa (z, t)| dt =

(25)

evolves in z according to
−α0 z

(30)

and

d
E(z)
dz

Theorem 3. Let q(z, t) ∈ L2 be a bounded signal propagating
according to (25), with brick-walls attenuation profile (5).
Then the signal energy
E(z) =

F̂ qa (z, t)

(31)
Consider now the nonlinearity and attenuation steps:
i
h
α0
2
(32)
qa (z, t) = e− 2 ∆z h(t) ∗ q(z, t)ejγ∆z|q(z,t)|

where a(t) is the inverse Fourier transform of α(ω).

Z∞

ω2

−∞

E(z) =

Consider next the time domain approach. Adding the frequency dependence of α to (1), we obtain

β2
2

where qa (z, t) , e∆zÂ e∆zN̂ q(z, t), and F̂ applies a Fourier
transform. This step causes only phase shifts in the frequency
domain which conserve the energy:
Z∞
Z∞
2
E(z + ∆z) =
|q(z + ∆z, t)| dt =
|qa (z, t)|2 dt.

B. Energy Conservation in the Time Domain

∂
1
β2 ∂ 2
q(z, t)
q(z, t) = − a(t) ∗ q(z, t) − j
∂z
2
2 ∂t2
2
+ jγ |q(z, t)| q(z, t)

(29)

∗

q(z, t2 )h(t − t2 ) dt2  dt1 dt.

(36)

In (35) we used the boundedness of q(z, t) to ensure that the
integrand is bounded by an integrable function, and therefore
the limit operator can be moved inside the integrals [6, Thm.
10.21]. The integral in square brackets in (36) is h(t) ∗ q(z, t).

4

where the convolutions are in ω3 .
α(ω) [dB/km]

Proof. The proof follows the same steps as the proof of
Theorem 1.
Theorem 4 implies that the energy per channel is not
necessarily conserved, even if α0 = 0.
B. Example: Three-Tone System
To illustrate the implications of Theorem 4, suppose we use
N = 3 vanishingly thin channels (tones) with spacing ∆ω:

α0
0

W1

W2

W3

W4

W5

ω [2π · Hz]

α0 = 0; ω1 = −∆ω; ω2 = 0; ω3 = ∆ω; Wn → 0.

Figure 1. Brick-walls attenuation profile.

The three-tone system is the limit of three modulated rectangular pulses when their duration T goes to infinity:

As q(z, t) is inside the passband W of the brick-walls filter
h(t), the result of this convolution is exactly q(z, t). Using
h(t) = h∗ (−t), we obtain
I1 = jγ

= jγ

Z∞

−∞
Z∞
−∞

q(z, t1 ) |q(z, t1 )|

2

Z∞

−∞

h∗ (t1 − t)q ∗ (z, t) dt dt1

|q(z, t1 )|4 dt1 .

(37)

Substituting (37) into (35) proves (27).

Q(z, ω) = lim 2πQ1 (z)δT (ω + ∆ω) + 2πQ2 (z)δT (ω)
T →∞

+ 2πQ3 (z)δT (ω − ∆ω)



T
T
sinc
ω .
(42)
δT (ω) =
2π
2π
R∞
2
We have δT (ω)∗δT (ω) = δT (ω), −∞ |δT (ω)| dω = T /(2π),
and we compute the power of the tones to be
PWn (z) = lim

Remark 1. The preservation of energy suggests that capacity
should grow with power. However, the variable E(z) is defined
based on an infinite time interval, allowing only one channel
use. We thus require a more precise result on the energy in a
finite time interval to be able to make a statement about the
channel capacity.
FLUCTUATIONS BETWEEN CHANNELS

A. General Expression for Energy Fluctuations
We have so far established energy conservation in a system
with brick-walls attenuation profile (5) and α0 = 0. Figure 1
shows α(ω) for a 5-channel system, where the attenuation is
α0 in the bands Wn , and ∞ outside them. We now characterize
the energy per WDM channel. Consider the signal
Qn (z, ω) = Hn (ω)Q(z, ω)

(38)

in channel Wn , where Hn (ω) , HWn (ω).

Theorem 4. Let Q(z, ω) ∈ L2 be a bounded frequencydomain signal propagating according to (2) with the brickwalls attenuation profile (5). Then the energy EWn (z) in
channel Wn , n ∈ {1, . . . , N }, evolves in z according to


γ
d
EWn (z) = −α0 EWn (z) − 3 ℑ
dz
4π 

Z∞

∗
[Q(z, ω3 ) ∗ Q(z, ω3 )] · [Qn (z, ω3 ) ∗ Q(z, ω3 )] dω3

−∞

(39)

(41)

where

T →∞

IV. E NERGY

(40)

EWn (z)
2
= |Qn (z)| .
T

(43)

Substituting (41) in (2), we obtain the FWM equations for the
evolution of each tone. For ω1 we have
h
β2
d
2
Q1 (z) = −j Q1 (z) + jγ |Q1 (z)| Q1 (z)
dz 
2

i
2
2
+2 |Q2 (z)| + |Q3 (z)| Q1 (z) + Q1 (z)Q∗2 (z)Q3 (z) .
(44)
Multiplying (44) by Q∗1 (z) and taking real parts, we have
n
d
4
2
2
PW1 (z) = −2γℑ |Q1 (z)| + 2 |Q2 (z)| |Q1 (z)|
dz
o
+2 |Q3 (z)|2 |Q1 (z)|2 + Q∗1 (z)Q2 (z)2 Q∗3 (z)

= −2γℑ Q∗1 (z)Q2 (z)2 Q∗3 (z) .
(45)
Only the FWM term Q∗1 (z)Q2 (z)2 Q∗3 (z) affects the signals’
4
powers. The self-phase modulation (SPM) term |Q1 (z)| and
2
the cross-phase modulation (XPM) terms 2 |Q2 (z)| |Q1 (z)|2
and 2 |Q3 (z)|2 |Q1 (z)|2 affect only the signals’ phases. The
power evolutions for the other channels are

d
PW2 (z) = −4γℑ Q1 (z)Q∗2 (z)2 Q3 (z)
dz

d
PW3 (z) = −2γℑ Q∗1 (z)Q2 (z)2 Q∗3 (z) .
dz

(46)

We see that the total power of the system is conserved:
d [PW1 (z) + PW2 (z) + PW3 (z)] /dz = 0.

(47)

Remark 2. In a two-tone system, the power in each channel
is conserved. This can be shown by setting Q3 (z) = 0.

5

C. Conditions for Per-Channel Energy Conservation
Definition 1. A system with attenuation profile (5) is energydecoupled if, for any bounded launch signal Q(0, ω) ∈ L2
and for each channel n, the evolution of its energy EWn (z) is
the same as if only that channel were present in the system:
EWn (z) = EWn (0)e−α0 z

∀Q(0, ω), ∀n.

(48)

Otherwise, the system is said to be energy coupled.
Equation (45) suggests that energy coupling is caused by
the FWM terms. This motivates us to try to avoid FWM in
order to ensure energy decoupling, as also done in [2]. We use
the standard definition of the sum of sets:
Wn1 +Wn2 = {ωn1 + ωn2 : ωn1 ∈ Wn1 , ωn2 ∈ Wn2 } . (49)
Lemma 5. Let W be a multi-channel frequency band as
defined in (7). Then the NLSE system (2) with attenuation
profile (5) is energy-decoupled (48) if and only if
(Wn1 + Wn2 ) ∩ (Wn + Wn3 ) = ∅, ∀ {n1 , n2 } 6= {n, n3 } .
(50)
The notation ∀ {n1 , n2 } =
6 {n, n3 } allows any n1 , n2 , n and
n3 in {1, . . . , N } such that (n1 , n2 ) 6= (n, n3 ) and (n1 , n2 ) 6=
(n3 , n), where (a, b) denotes an ordered pair. For example,
(n1 , n2 ) = (1, 1),

(n, n3 ) = (1, 2)

(51)

(n, n3 ) = (2, 1)

(52)

is allowed, but
(n1 , n2 ) = (1, 2),
is not.
PN
Proof. Substituting Q(z, ω) = n′ =1 Qn′ (z, ω) in (39), the
condition to prevent energy coupling is
( N N N
)
X X X
ℑ
In1 n2 nn3 = 0, ∀n ∈ {1, . . . , N }
n1 =1 n2 =1 n3 =1

(53)

The sum (53) has 64 terms. We call In1 n2 mn3 an overlapping
term if ωn1 + ωn2 = ωm + ωn3 , and a non-overlapping term
otherwise. Non-overlapping terms are equal to 0. We consider
groups of terms In1 n2 mn3 with nonnegative imaginary part.
• Case 1) The 27 terms of the form In1 n2 mn3 , where
n1 , n2 , n3 6= m: these terms are equal to exp (jπ/8) if
they are overlapping, with positive imaginary part.
• Case 2) The 9 terms of the form In1 n2 mm where n1 , n2 6=
m and n3 = m: these terms are equal to exp (jπ/4) if
they are overlapping, with positive imaginary part.
• Case 3) The 18 terms of the form Imn2 mn3 or In1 mmn3
where n3 6= m, and either n1 = m or n2 = m: these
terms are equal to 1 if they are overlapping.
• Case 4) The 9 terms where only one of n1 , n2 , n3 is not
equal to m: these terms form the following groups:
Immmn3 + Imn3 mm + In3 mmm = 2In3 mmm + In∗3 mmm
where In3 mmm = exp (jπ/8) if it is overlapping. The
imaginary part is again positive.
• Case 5) The term Immmm = 1.
This list shows that all the terms of the sum (53) have either
positive or zero imaginary part. If any term in Cases 1, 2 or 4
is overlapping, then the sum has positive imaginary part and
the proof is complete. Case 5 is not included in (50). If only
terms of Case 3 are overlapping, then we have an overlapping
term Imn2 mn3 with n2 6= n3 (if n2 = n3 , the term is again
not included in (50)). But then In2 mn3 m is an overlapping
term of Case 1, and thus (53) is not satisfied for n = n3 . This
completes the proof: if any overlapping term exists, then the
per-channel energy is not conserved.
Remark 3. Condition (50) in a system with distributed filtering ensures the absence of FWM, and thus ensures per-channel
energy conservation. However, interaction between channels
still occurs due to XPM.

where
In1 n2 nn3 ,

Z

D. Spectral Efficiency of an Energy-Decoupled System

∞

−∞

[Qn1 (z, ω) ∗ Qn2 (z, ω)]
∗

· [Qn (z, ω) ∗ Qn3 (z, ω)] dω. (54)

The terms with {n1 , n2 } = {n, n3 } are real-valued. If (50) is
true, then all the other terms are products of convolutions that
do not overlap in frequency, which implies (48).
To prove that (48) implies (50), suppose that (50) is false,
i.e., there is a choice of {m1 , m2 } 6= {m, m3 } for which
(Wm1 + Wm2 ) ∩ (Wm + Wm3 ) 6= ∅.

(55)

Choose ωm1 ∈ Wm1 , ωm2 ∈ Wm2 , ωm ∈ Wm , ωm3 ∈ Wm3 ,
such that
(56)
ω m1 + ω m2 = ω m + ω m3 .
Now consider n = m in (53), and choose the following fourtone launch signal:

if n∈{m1 ,m2 ,m3 }

δ (ω − ωn ) ,
and n6=m
Qn (0, ω) = e−jπ/8 δ (ω − ωn ) , if n = m
(57)


0,
otherwise.

Energy-decoupled systems have the potential to encode
information in the energy of each channel and communicate
without inter-channel interference. We develop an upper bound
on the spectral filling efficiency
PN
n=1 Wn
(58)
η(N ) =
max W − min W

of an energy-decoupled system. We consider only systems
with equal channel widths Wn = W, ∀n. Suppose ω 1 =
0.5W , and let mn = 1 + (ω n − 0.5W )/(2W ), so that we
have
Wn = [(2mn − 2)W, (2mn − 1)W ] ,

mn ≥ 1.

(59)

From (50) and (59), the condition for energy-decoupling is
6 {n, n3 }
|(mn1 + mn2 ) − (mn + mn3 )| ≥ 1, ∀ {n1 , n2 } =
(60)
i.e., all pairwise sums of mn differ by at least 1. This condition
was derived in [2] to ensure the absence of FWM. The case

6

where the mn are constrained to be integers was also addressed
in [2]; in this case (mn ) is a Sidon sequence [7] which satisfies
mn1 + mn2 6= mn + mn3 , ∀ {n1 , n2 } 6= {n, n3 } .

(61)

In an energy-decoupled system, the mn in (59) must form a
sequence of real numbers satisfying (60). We call this a RSidon sequence.
The question remains how efficiently the spectrum can be
filled with this method. For integer k, let N (k) be the length
of the longest R-Sidon sequence with elements in {1, . . . , k}:
(
mn ∈ {1, . . . , k} ∀n ∈ {1, . . . , N }
N (k) = max N s.t.
(mn )
(mn )N
n=1 is a R−Sidon sequence.
(62)
Lemma 6.

N (k)
lim sup √ ≤ 1.
k
k→∞

(63)

Proof. Erdös proved (63) for Sidon sequences [8]. We extend
the proof to R-Sidon sequences. Consider the sequence (mn ),
with 1 ≤ m1 < . . . < mN ≤ k, and which satisfies (60).
Consider a positive integer a ≤ k, and define the intervals:
Iu , [u − a, u) ,

u ∈ {1, 2, . . . , k + a} .

(64)

Let Mu be the number of mi ’s in Iu . As each mi occurs in
a intervals, we have
k+a
X

Mu = N a.

(65)

u=1

Let Pu = Mu (Mu − 1) /2 be the number of pairs
(mi , mj ), i < j in interval Iu . Using (65) and
!2
k+a
k+a
X
X
1
2
Mu ≥
(66)
Mu
k + a u=1
u=1
we have
Np ,

k+a
X

u=1

Pu ≥

1
Na
2





Na
−1 .
k+a

(67)

For i < j, consider the differences di,j = mj − mi . By (60),
the distance between any two di,j must be least 1. Let r be
an integer. If there is one di,j in [r, r + 1), then set dr = di,j ;
if not, set dr = a. Note that dr ≥ r. Let Mi,j be the number
of intervals Iu that contain the pair (mi , mj ). We have
Np =

X

(i,j) : di,j ≤a

≤

a−1
X
r=1

a−1
(a) X

Mi,j ≤

(a + 1 − r) =

r=1


 s 
2
k
k
(a + 2)(a − 1)(k + a)
1
1+
+
1+
+
.
a
4
a
a2
√


 (70)
Choosing a = k 3/4 , we have N = k + O k 3/8 , which
proves the lemma.
1
N≤
2

The asymptotic bound (63) is achieved by Bose’s construction of Sidon sequences [9]. When N is a prime power, let FN
be the finite field of size N , and θ a generator of the extended
field FN 2 (a root of an irreducible polynomial p(x) of degree
2 in FN ). For a set S, let (S) be the sequence that contains
the elements of S in increasing order. The Bose sequence



MBose (N ) , m ∈ 1, . . . , N 2 − 1 : θm − θ ∈ FN
(71)
is a Sidon sequence. For example, for N = 11 and p(x) =
x2 + x + 7, we choose θ = x and obtain
MBose (N ) = (1, 6, 22, 62, 68, 69, 71, 88, 99, 103, 113).
(72)
If N is not a prime power, then we use the next prime
power and truncate the resulting sequence. As N (k) =
|MBose (N )| = N [10, p. 597], and max {MBose (N )} ≤
N 2 − 1, Bose’s construction achieves the upper bound (63):
lim

k→∞

N
N (k)
√ = lim p
= 1.
N →∞
k
max MBose (N )

(73)

Remark 4. Integer-valued Bose sequences achieve the asymptotic upper bound (63). Therefore, the mn are constrained to
be integers in the following.
Remark 5. For small N , there are Sidon sequences with
smaller k than the Bose sequences. The optimal sequences
for N ≤ 16 are listed in [11] and references therein.
At this point, we are ready to bound the maximum spectral
filling efficiency of an energy-decoupled system.
Theorem 7. Let W be a multi-channel frequency band, as
defined in (7), with constant channel widths Wn = W .
Furthermore, let W be such that the NLSE system (2) with
attenuation profile (5) is energy-decoupled, i.e., W fulfills (50).
Then the optimal spectral filling efficiency (58) for N channels
belongs to O(1/N ) and we have
lim η(N )N =

N →∞

1
.
2

(74)

Proof. The spectral filling efficiency of (59) is

⌈a − dr ⌉

1
(a + 2)(a − 1)
2

which implies

η(N (k)) =
(68)

where (a) holds because, for di,j ≤ a, we have Mi,j ≤
⌈a − di,j ⌉. Combining (67) with (68), we obtain


(a + 2)(a − 1)
Na
−1 ≤
(69)
N
k+a
a

N (k)W
.
(2k − 1)W

(75)

Multiplying by N (k) and using (73), we obtain (74).
Theorem 7 proves that a N -channel energy-decoupled system with uniform channel width W can asymptotically fill
at most a fraction 1/(2N ) of the spectrum. A system with
unequal channel widths might be able to achieve better spectral
efficiency. For N = 2 and N = 3, standard optimization

7

·10−12

·10−12

4

4
EW1 (z)
EW4 (z)

EW2 (z)
EW5 (z)

EW (z)
EW3 (z)

E(J)

E(J)

EW (z)
EW3 (z)

2

0

0

20

40

60

80

100

120

140

160

z(km)

EW1 (z)
EW4 (z)

EW2 (z)
EW5 (z)

2

0

0

20

40

60

80

100

120

140

160

z(km)

Figure 2. Energy evolution in a 5-channel system using a Sidon sequence
and filters every 10 km.

Figure 3. Energy evolution in a 5-channel system with uniform channel
spacing in frequency and filters every 10 km.

techniques show that equal width channels maximize the
spectral efficiency. For N > 3, the largest channel width must
be less than or equal to the smallest empty space between
channels (to see this, let n1 = n be the largest channel, and
n2 and n3 be the two closest channels in (50)). Systems with
equal channel widths have all widths equal to the minimum
spacing. This does not guarantee optimal spectral efficiency,
but leads us to believe that allowing unequal widths gives small
improvements only.

the new model preserves energy, thus establishing an invariant
that can be useful for communications.
We have characterized the evolution of the per-channel
energy in a WDM system with distributed filtering. We have
derived conditions that ensure per-channel energy conservation
by using Sidon sequences. For constant channel widths, we
have proved that an N -channel Sidon system can asymptotically fill at most a fraction 1/(2N ) of the spectrum, which
implies that spectral efficiency goes to 0 as N increases. The
capacity analysis of this new band-limited channel model is
an interesting open problem.

E. Numerical Results
Consider a 5-channel system with dispersion coefficient,
β2 = −21.667 ps2 /km, nonlinear coefficient γ =
1.2578 W−1 km−1 , and channel bandwidth W/(2π) =
1 GHz. To demonstrate the practical use of the scheme, we
drop the assumption of ideal distributed filtering and use brickwalls filters (6) every ∆z = 10 km. In a first numerical
experiment, the channels are placed according to the densest
Sidon sequence: (1, 2, 5, 10, 12), i.e., their centers are at
ω n = (0.5W, 2.5W, 8.5W, 18.5W, 22.5W ) .

(76)

In a second experiment, the channels are uniformly spaced
in frequency in the same bandwidth of 23W . A scaled root
raised cosine pulse with roll-off factor of β = 0.15 and
total bandwidth W is sent in each channel. The five scaling
factors are chosen randomly to have the respective phases
{−2.397, −0.217, 2.065, 2.937, 3.003} rad and pulse energies
{0.039, 0.468, 1.469, 1.160, 1.166} pJ.
Due to the finite filter spacing, some of the launch energy
is lost during propagation. In the Sidon system (Figure 2),
2.2% of the energy is lost, but the energy per channel stays
approximately constant. In the uniform system (Figure 3), the
energy loss is 0.98%, but the energy fluctuations between
channels are apparent.
V. C ONCLUSION
We have proposed a distributed filtering approach to mitigate spectral broadening in the NLSE. We have proved that

VI. ACKNOWLEDGMENT
The authors would like to thank R.-J. Essiambre, A.
Mecozzi, and M. Shtaif for stimulating discussions on energy
conservation in fiber with bandwidth constraints.
R EFERENCES
[1] R. J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel,
“Capacity Limits of Optical Fiber Networks,” J. Lightw. Technol.,
vol. 28, no. 4, pp. 662–701, Feb 2010.
[2] F. Forghieri, R. W. Tkach, and A. R. Chraplyvy, “WDM systems with
unequally spaced channels,” J. Lightw. Technol., vol. 13, no. 5, pp. 889–
897, May 1995.
[3] G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. Academic Press, October
2012.
[4] E. Titchmarsh, Introduction to the Theory of Fourier Integrals. Clarendon Press, 1948.
[5] J. Szarski, Differential inequalities, ser. Monografie matematyczne.
Państwowe Wydawn. Naukowe, 1967.
[6] A. Browder, Mathematical Analysis: An Introduction. Springer New
York, 1996.
[7] S. Sidon, “Ein Satz über trigonometrische Polynome und seine Anwendung in der Theorie der Fourier-Reihen,” Math. Annal., vol. 106, pp.
DXXXVI–DXXXIX, 1932.
[8] P. Erdös and P. Turán, “On a Problem of Sidon in Additive Number
Theory, and on some Related Problems,” J. London Math. Soc., vol.
s1-16, no. 4, pp. 212–215, 1941.
[9] R. C. Bose, “An affine analogue of Singer’s theorem,” J. Indian Math.
Soc. (N.S.), vol. 6, pp. 1–15, 1942.
[10] G. Martin and K. O’Bryant, “Constructions of generalized Sidon sets,”
J. Combin. Theory, Ser. A, vol. 113, no. 4, pp. 591 – 607, 2006.
[11] J. B. Shearer, “Some new optimum Golomb rulers,” IEEE Trans. Inf.
Theory, vol. 36, no. 1, pp. 183–184, Jan 1990.

