Dispersion-Managed Solitons in the Path-Average
Normal Dispersion Regime
by
Samuel Tin Bo Wong
Submitted to the Department of Electrical Engineering and Computer
Science
in partial fulfillment of the requirements for the degree of
Master of Engineering in Electrical Engineering and Computer Science
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
May 2001 O
® Samuel Tin Bo Wong, MMI. All rights reserved.
The author hereby grants to MIT permission to reproduce and
distribute publicly paper and electronic copies of this thesis document BARKER
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Department of Electrica Engineering and Computer Science
May 23, 2001
Certified by ..............................
C ertified by .....
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A ccepted by .............
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Hermann A. Haus
Institute Professor
Thesis Supervisor
.............................
Scott A. Hamilton
MIT Lincoln Laboratory Staff
4hes ,*Supervisor
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....................
Arthur C. Smith
Chairman, Department' Committee on Graduate Students
Dispersion-Managed Solitons in the Path-Average Normal
Dispersion Regime
by
Samuel Tin Bo Wong
Submitted to the Department of Electrical Engineering and Computer Science
on May 23, 2001, in partial fulfillment of the
requirements for the degree of
Master of Engineering in Electrical Engineering and Computer Science
Abstract
Optical fiber systems offer high-speed, broadband, and long-haul communications.
Solitary waves known as optical solitons seem to be a natural means of transmitting
data via fiber because of the balancing effects of linear anomalous group velocity
dispersion (GVD) and nonlinear intensity-dependent self-phase modulation (SPM).
Regular soliton systems (with uniform anomalous fiber), however, have yet to enter
commercial markets because of high power requirements and timing jitter. Another
class of solitary waves called dispersion-managed (DM) solitons, which can be manifested via a dispersion map, can resolve these issues. Numerical simulations and
variational methods have shown that DM solitons can propagate in the path-average
anomalous, normal, and zero dispersion regimes because of the interplay involving
dispersion, nonlinearity, and chirp. In the net normal dispersion regime, a plot of
pulse energy versus net dispersion yields two theoretical energy branches with map
strength as a parameter. The lower-energy branch is interesting since it lies on a
"quasi-linear" region near net zero dispersion. Having a net dispersion around zero
means reduced timing jitter. The lower-energy DM solitons also need less power but
exploit enough nonlinearity to fight GVD. So far, no one has claimed to have actually found these pulses. This thesis provides preliminary experimental evidence for
the existence of these lower-energy DM solitons. Numerical and experimental studies
show a shifting of the transform-limited state position due to a dispersion imbalance.
Under the proper initial launch conditions, nonlinearity can mitigate the effects of
this dispersion-induced shifting in order to produce periodically stationary pulses.
These DM solitons can potentially gain a edge over linear techniques used today.
Thesis Supervisor: Hermann A. Haus
Title: Institute Professor
Thesis Supervisor: Scott A. Hamilton
Title: MIT Lincoln Laboratory Staff
2
Acknowledgments
I would first like to thank Prof. Hermann Haus for his guidance and mentorship
throughout my thesis project. I am forever grateful not only because he introduced
me to an exciting area of research but also because he gave me some much-needed
direction when my Lincoln group was suffering a mass exodus with people leaving for
optical startup companies at that time. Like every one of his students before me, I
am very honored to have Prof. Haus as my advisor. I would like to very much thank
Scott Hamilton for acting as my Lincoln supervisor despite his having many other
time-consuming duties in the Lab. I greatly appreciate his experimental advice and
careful reading of my thesis. And, of course, I owe my skiing experiences to Scott.
There are also various other people I must thank. I thank Jeff Minch, my previous
Lincoln supervisor, for showing me how to construct a recirculating fiber loop. I thank
John Moores for answering my (naive) questions on dispersion-managed solitons when
I first started to study them. I thank Prof. Erich Ippen for taking some of his valuable
time to discuss my experimental setup and results and providing some very helpful
insight (I wished I had more opportunities to talk with him). I thank Tom Murphy
for his gracious help in the dispersion measurements and numerical simulations. I
thank Leaf Jiang, my officemate on campus, for expressing interest in my thesis and
for helping me on numerous occasions. I thank Bryan Robinson, the "Senior Staff"
of the TDM lab, for giving me occasional experimental help and also Shelby Savage
for letting me time-share the modelocked fiber laser (and providing some data for
the laser). And I would like to thank Todd Ulmer for being cool and for introducing
Van Halen to me so I can add punk, er, rock music to my "high-brow" repertoire of
Bach, Beethoven, and Mozart (I need my daily dosage of "Hot for Teacher" to prep
me up for labwork). Finally, I wish to extend my thanks to anyone in my Lincoln
and campus research groups who had helped me in any little way.
I suppose at this point I need to give the obligatory thanks to my family like any
acknowledgments section of a typical dissertation. Well, I'm not going to do that not
because I don't care about them but because it seems so cliche. I know my family
supports me no matter what endeavour I undertake and for me to openly thank them
would be superfluous. Sincerity is all that matters. I would, however, like to take this
opportunity to thank my support network of friends at MIT, especially those whom
I met during my senior year. I guess I should also acknowledge someone at MIT (she
knows who she is) for allowing me to realize that there's so much more to life than
just math and physics. But I'm not going to talk about my philosophy on life right
now. I shall reserve that for the acknowledgments section of my doctoral thesis.
3
Contents
1
Introduction
10
2
Background on Fiber Properties and Regular Solitons
13
Intrinsic Fiber Properties . . . . . . . . . . . . . . . . . . . . .
13
2.1.1
D ispersion . . . . . . . . . . . . . . . . . . . . . . . . .
13
2.1.2
Frequency Chirp
. . . . . . . . . . . . . . . . . . . . .
15
2.1.3
Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . .
16
2.1.4
Attenuation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
2.2
Wave Propagation in a Nonlinear Medium . . . . . . . . . . . . . . .
19
2.3
Discussion on Regular Solitons . . . . . . . . . . . . . . . . . . . . . .
22
2.1
3
2.3.1
The Nonlinear Schrbdinger Equation
. . . . . . . . . . . . . .
22
2.3.2
Properties of Regular Solitons . . . . . . . . . . . . . . . . . .
26
2.3.3
Limiting Factors for Regular Soliton Optical Networks
. . . .
28
Background on Dispersion-Managed Solitons
31
3.1
Introduction to Dispersion-Managed Solitons . . . . . . . . . .
31
3.2
Methods for Theoretical Analysis . . . . . . . . . . . . . . . .
35
3.2.1
Numerical Simulation: The Split-Step Fourier Method
35
3.2.2
Approximate Method: The Variational Approach
3.3
. . .
39
Behavior and Characteristics of DM Solitons . . . . . . . . . .
44
4 Experimental Search for Lower-Energy DM Solitons
4.1
Experimental Objective
. . . . . . . . . . . . . . . . . . . . . . . . .
4
48
49
4.2
4.3
Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
4.2.1
Laser Source . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
4.2.2
Dispersion Map and Measurements . . . . . . . . . . . . . . .
51
4.2.3
Other Measurements . . . . . . . . . . . . . . . . . . . . . . .
57
4.2.4
Recirculating Fiber Loop . . . . . . . . . . . . . . . . . . . . .
60
Experimental Results and Discussion . . . . . . . . . . . . . . . . . .
61
4.3.1
Preliminary Observed Effects of Power Level . . . . . . . . . .
62
4.3.2
Shifting of the Minimum Pulse Width Position . . . . . . . . .
66
4.3.3
Robustness of Pulses After Long-Distance Propagation . . . .
69
4.3.4
Achieving Periodically Stationary Pulses . . . . . . . . . . . .
70
4.3.5
Discussion of Experimental Results . . . . . . . . . . . . . . .
73
5 Conclusion and Future Work
76
A Numerical Simulation of Experiments
79
5
List of Figures
2-1
Diagram of anomalous group velocity dispersion and Kerr nonlinearity
(self-phase modulation). Proper balancing between these two effects
induces pulse shape stabilization for soliton propagation. . . . . . . .
2-2
Comparison between NRZ and soliton transmission formats for the
given data stream . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-1
32
Numerical simulation of DM soliton in one unit cell of simple two-stage
dispersion m ap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-3
30
Example of a symmetric two-stage dispersion map with a path-average
anomalous dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-2
27
33
Variational plot of DM soliton energy versus net normal dispersion
with pulse width as a parameter (provided by Prof. Haus). The circles
on the plot represent direct numerical simulations. . . . . . . . . . . .
4-1
45
Autocorrelation (left) and optical spectrum (right) of the transmitter
laser pulse. The autocorrelation FWHM translates to At = 2.5 ps
and the bandwidth tranlates to Av = 160 GHz for a time-bandwidth
product of about 0.4, which is close to the Gaussian transform-limited
state. The pedestal in the spectrum is ASE noise produced by an
optical am plifier.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
. .
52
4-2
Experimental setup for dispersion measurements of fiber sections.
4-3
Group delay measurements (top) and dispersion calculation (bottom)
for first anomalous segment consisting of 50 km AllWaveTM . . . . . .
6
54
4-4
Group delay measurements (top) and dispersion calculations (bottom)
for second anomalous segment consisting of 25 km AllWaveTM
4-5
Group delay measurements (top) and dispersion calculations (bottom)
for third anomalous segment consisting of 10 km SMF.
4-6
. . . . . . . .
55
Group delay measurements (top) and dispersion calculations (bottom)
for normal segment consisting of 15 km DCF.
4-7
55
56
. . . . . . . . . . . . .
Group delay (top) and GVD (bottom) calculations for entire fiber loop
consisting of 75 km AllWave T M , 10 km SMF, and 15 km DCF. Net
zero dispersion is achieved at 1550 nm as designed.
4-8
4-9
. . . . . . . . . .
56
Experimental setup for EDFA transfer function measurements with
amplifier pump level varied between 1 and 10. . . . . . . . . . . . . .
58
Measured amplifier gain and saturation power at different pump levels.
59
4-10 Experimental setup of recirculating fiber loop
. . . . . . . . . . . . .
60
4-11 Dispersion map with a tap at launch point located 7.5 km before the
anomalous half-cell center. . . . . . . . . . . . . . . . . . . . . . . . .
63
4-12 Experimental data of pulse width evolution versus loop period measured at the launch point in the fiber loop. . . . . . . . . . . . . . . .
65
4-13 Illustration of how the tranform-limited state position shifts in the
linear case with negligible loss and nonlinear effects. . . . . . . . . . .
66
4-14 Autocorrelation of pulses after 3000 km propagation for "low" (left)
and "high" (right) power levels with 6.7 km SMF anomalous loop compensation fiber. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
4-15 Dispersion map with launch point located 7.5 km after the anomalous
half-cell center and a tap located 1.7 km from the launch point.
. . .
71
4-16 Experimental data of pulse width evolution versus loop period mea72
sured at the loop tap point . . . . . . . . . . . . . . . . . . . . . . . .
A-1 Simulation of the linear case with only dispersion and no nonlinearity
or loss........
...............
...........
7
........
.81
A-2 Simulation of fiber loop with enough amplifier gain to compensate fiber
lo ss.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
A-3 Simulation of fiber loop with a higher pump level to show that nonlinearity can mitigate the shifting of the minimum pulse width position.
8
82
List of Tables
4.1
Measured loss for each fiber section in loop.
. . . . . . . . . . . . . .
57
4.2
OTDR measured length for each fiber section in loop. . . . . . . . . .
58
9
Chapter 1
Introduction
Optical fiber communications technology seems very promising in developing net-
works that provide higher bandwidth, faster speeds, and more stable transmission of
information. Essentially, optical fiber networks are the key to a better Internet. Issues
exist, however, concerning the implementation of such systems due to the intrinsic
physical properties of optical fiber. One major effect is dispersion, which causes temporal broadening of the optical pulse envelope as it propagates along fibers. Another
well-known source of distortion is the Kerr-induced nonlinearity, which is significant
if the intensity of the electric field of the optical pulse is sufficiently high.
Despite the inherent dispersive and nonlinear nature of optical fibers, it is possible
to exploit these two apparently detrimental effects to create a stable, "particle-like"
pulse, known as a soliton. Soliton systems use these fiber properties to their advantage
by compensating broadening due to dispersion with the compression imposed by
nonlinearity. Solitons seem to be ideally suited for the transmission of data in longdistance and high-speed communication systems.
However, despite the potential of solitons to balance nonlinearity and dispersion
in fiber, no known commercial soliton system is currently deployed [1].
One of the
primary reasons is that solitons suffer from jitter, a pulse-position modulation distortion that is dependent on dispersion and can be caused by a number of sources. The
most well-known of such timing jitters is the Gordon-Haus effect, which results from
amplifier noise. In addition, regular solitons exist only in the anomalous (negative)
10
dispersion regime. Propagation conditions require proportionality between soliton
power and dispersion in the fiber. Solitons, therefore, cannot tolerate very low dispersion (around zero dispersion) because the soliton power, along with the signal
power, would vanish. This condition means that decreasing dispersion in order to
reduce jitter comes at the expense of the signal-to-noise ratio. An additional disadvantage for regular soliton systems is the high power required to support soliton
propagation. High power levels are needed to induce sufficient nonlinearity to counter
the high dispersion in optical fibers. In principle, regular solitons can exist with low
energies if only fibers can be reliably manufactured with uniform low dispersion. This,
however, is a fabrication problem because not only is it difficult to make fibers with
low dispersion, fluctuations in the regime of low dispersion due to the manufacturing
process may severely perturb the stable dynamics of a soliton pulse. From a commercial perspective, since linear techniques, whose philosophy is to fight or suppress
nonlinearity as opposed to soliton methods, seem to perform quite well under the
current circumstances (with data rates around 10 Gbit/s), there is no practical or
desirable incentive to implement soliton systems by using higher power levels.
Dispersion-managed solitons address the issues described above and show promise
in overcoming the limitations of regular solitons [1]. Dispersion-managed (DM) solitons occur in systems with fiber having spatially varying dispersion that is usually
periodic. The advantages of such solitons over their conventional counterparts include
enhanced energy, which leads to reduced Gordon-Haus jitter, and reduced pulse interaction, which leads to higher bandwidth efficiency [1]. Theoretical studies of fiber
maps with segments of varying dispersion reveal that DM solitons can propagate
at zero or normal (positive) average dispersion, whereas ordinary solitons propagating through fiber of constant dispersion strictly operate in the anomalous dispersion
regime.
The focus of this M.Eng. thesis is to investigate DM soliton propagation in the
net normal dispersion regime near zero dispersion. Previous theoretical studies [2,
3, 4, 5, 6] have shown that two pulse energy solutions exist for a fixed net normal
dispersion with pulse width and map strength as parameters, provided that the map
11
strength is sufficiently strong. The higher-energy solutions are not surprising since
they are natural extensions of regular solitons; but, the lower-energy solutions are
unique to DM soliton propagation in the net normal dispersion regime and have
not been thoroughly investigated. The lower-energy DM solitons may have reduced
optical power requirements comparable to systems employing linear techniques and
so the implementation of DM solitons may be more attractive than regular solitons.
Research on this "quasi-linear" lower energy branch may prove to be quite interesting
if these lower-energy dispersion-managed solitons can be experimentally demonstrated
to exist in the net normal dispersion regime.
The layout of this report is divided into five chapters.
Chapter 2 gives brief
background information necessary to understand the unique class of solitary waves
called dispersion-managed solitons.
In this chapter, various properties of optical
fiber are discussed, along with the concept of the regular optical soliton. Chapter 3
provides some past preliminary theoretical and numerical studies on DM solitons in
order to give motivation for this project. Chapter 4 presents the experimental work
done for this thesis. The experimental setup, procedures, and results are discussed
to demonstrate preliminary evidence for the existence of lower-energy DM solitary
waves. Chapter 5 ends this dissertation with a conclusion and plans for future work.
12
Chapter 2
Background on Fiber Properties
and Regular Solitons
This chapter introduces some basic concepts that are helpful in understanding dispersion-managed solitons. Wave propagation through fiber and material properties, such
as dispersion, nonlinearity, and attenuation, will be discussed. A brief introduction
to regular solitons, which requires constant anomalous dispersion, will be presented.
A key tool discussed in this section is the Nonlinear Schr6dinger Equation (NLSE),
which is one of the simplest ways to model nonlinear pulse propagation. A section
on the implications of regular soliton implementation is also included.
2.1
2.1.1
Intrinsic Fiber Properties
Dispersion
The effect of dispersion on a pulse is temporal broadening. Consider the propagation
constant (or wave number)
# expanded
in a Taylor series around an angular frequency
WO
1
O(wo) =
0 + 0 1 (W - wO) + -0 2 (W
1
-
13
3
2
W
0 ) + -0 3 (W - WO) +
...
,
(2.1)
where
#4 =
The first two terms, #3
(vp) and group (v_)
(2.2)
.
,
wo/v, and i1 = 1/v, in Eq. (2.1), are related to the phase
velocities, respectively, and the third term
#2 represents
group-
velocity dispersion (GVD) or chromatic dispersion. GVD is the phenomenon of different frequencies (or wavelengths) in a pulse traveling at different velocities. Since some
optical frequencies under the pulse envelope propagate faster or slower than others,
the envelope broadens in time as it moves along a dispersive medium. Dispersion is a
linear phenomenon. According to the properties of Fourier transformation, a shorter
pulse in time implies a larger bandwidth in frequency. Because wide-bandwidth pulses
contain more frequency components traveling at different speeds due to dispersion,
short pulses broaden at a faster rate than long pulses. GVD-induced intersymbol
interference (ISI) is one challenge faced when implementing optical fiber networks
employing ultrashort pulses.
GVD is often represented in terms of wavelength rather than in angular frequency. In terms of wavelength, the chromatic dispersion parameter is renamed to
D = df31 /dA.
This definition can be related to
#2 in
Eq. (2.1) using the dispersion
relation as follows
D=d
dA
1 D
v9
o(
27rc(23
#2 .(2.3)
A213
The minus sign in the equation comes from the inverse relationship between wavelength and angular frequency, as indicated in the expression Aw = 27rc. Note that
the units for
/
is [ps 2 /km] while the units for D is [ps/nm -km].
The dispersion
parameter D can be physically interpreted as the spreading in time (in ps) per unit
bandwidth of the pulse (in nm) over one kilometer of fiber.
There are two types of GVD: anomalous (or negative) and normal (or positive)
dispersion.
The type is determined by the sign of the dispersion parameter.
14
A
negative 02 (or positive D) is defined as anomalous while a positive
/2
(or negative D)
is normal. In anomalous dispersion, higher frequencies (shorter wavelengths) travel
faster than lower frequencies (longer wavelengths). The converse is true in normal
dispersion. In standard single-mode fiber (SMF), the zero-dispersion wavelength is
approximately 1.3 pm with the anomalous dispersion regime spanning the longer
wavelength side and the normal dispersion regime spanning the shorter wavelength
side. Adjustment of the index of refraction profile and core dimensions in the fiber
changes the zero dispersion wavelength and the dispersion slope, i.e.
dispersion-
shifted fiber (DSF) or dispersion-compensated fiber (DCF). To first order, an equal
amount of anomalous dispersion completely compensates an equal amount of normal
dispersion and this case is the simplest example of dispersion management.
Higher-order dispersions exist in addition to GVD. Third-order dispersion (TOD)
or /3 gives rise to a dispersion slope (i.e. how GVD changes over frequency or wavelength) and creates an asymmetry in the dispersion profile. Polarization-mode dispersion (PMD) results from random polarization effects since fiber is a birefrigent
material. To first order, two eigenstates exist where the fast principal axis is orthogonal to the slow axis. When random polarization is induced on a propagating
light pulse, different components of the wave fall onto one of the axes depending on
their polarization. Since the components on different polarization axes travel at different velocities, the end result is temporal broadening of the pulse envelope. GVD
is generally the dominant dispersion component since its operational distance is the
greatest. Higher-order dispersions, such as PMD, however, become non-negligible at
higher transmission data rates.
2.1.2
Frequency Chirp
Frequency chirp refers to the time dependence of signal frequency. GVD induces linear
chirp on an optical pulse propagating through fiber. Positive chirp occurs when the
instantaneous frequency increases linearly from the leading to the trailing edge (also
known as up-chirp) and for negative chirp, the converse is true. If a pulse is initially
unchirped, its temporal width broadens by the same amount in either the anomalous
15
or normal dispersion regime. If a pulse is chirped, however, its behavior is different
in different dispersion regimes depending on the sign of the chirp.
A positively-
chirped pulse temporally broadens in anomalous dispersion, but compresses initially
followed by broadening in normal dispersion. Similarly, a negatively-chirped pulse
broadens in normal dispersion, but compresses initially followed by broadening in
anomalous dispersion. Chirped-pulse compression occurs because the GVD of one
sign is temporarily compensated by a pulse chirp of the opposite sign. It can be
inferred that anomalous dispersion generates positive chirp whereas normal dispersion
induces negative chirp. If an optical pulse has no chirp, the time-bandwidth product
AwAt is minimized and the pulse is transform-limited.
2.1.3
Nonlinearity
Nonlinearity in optical fiber becomes a significant effect if the intensity of the electric
field of the light pulse is sufficiently high. This fiber nonlinearity induces a change in
the index of refraction
n = no + n 2JE12
(2.4)
,
where no is the bulk material index and n 2 is the new constant that determines how
the index increases with increasing optical intensity. This is known as Kerr-induced
nonlinearity. The dipole moment per unit volume, or polarization P, can be written
in a power series in terms of electric field E:
P = O(XN E +
X(2)E2 + X(3)E+...)
.
(2.5)
co is the permittivity (or dielectric constant) in free-space and X(n) is an nth-order
tensor representing the nonlinear optical susceptibility. Note that P and E are space,
time, and frequency dependent. The first term in the series is the linear polarization
16
and the rest of the terms comprise the nonlinear polarization, that is,
PNL =EoX(2) E
and
PL =OX('E
2
+ X( 3)E 3
+...)
(2.6)
.
In this analysis, only the dominant term in the nonlinear polarization is retained.
For optical fiber, X( 2 ) is negligible due to the inversion symmetry of the potential
around the SiO 2 molecules. We can therefore describe the nonlinear polarization of
a propagating beam with angular frequency w in optical fiber as
PNL(W)
=
3oXo(
3
)(P
: W, -w, w)E(w)12 E(w)
(2.7)
,
where the coefficient of 3 accounts for the number of possible permutations for the
electric field orientation (the degeneracy factor). The total polarization of the material
system is now written as
Ptt = co(X(01 E(w) + 3x(3 (W : w, -w, w)IE(w)| 2 E(w)) = OXeffE(w)
,
(2.8)
with the effective susceptibility defined as
Xef f = X(1 + 3X(3 (W : w, -w, w) IE(w) 12
.
(2.9)
It is generally true that [7]
n 2 = 1 + Xef f
(2.10)
and combined with the intensity-dependent index of refraction from Eq. (2.4), the
linear and nonlinear indices can be related to the linear and nonlinear susceptibilities
in the following manner:
no = (1 + X(l)) 1/ 2
17
(2.11)
and
n2 =
8no
.X
(2.12)
In a single optical fiber channel, the dominant Kerr nonlinearity described by X( 3)
is self-phase modulation (SPM). SPM is a change in the phase of the optical pulse
caused by the nonlinear index of refraction (a change in the index modifies the phase
velocity). As described by its name, SPM is a modulation of the pulse phase due to
its own intensity. This phase delay is proportional to the intensity of the pulse. The
shift in frequency (Aw) is then the time derivative of the change in phase (Aq), i.e.
Aw-
dt
dt
(2.13)
Unlike linear processes, nonlinearity can modify the frequency content of a pulse
and even generate new spectral components. In the case of SPM, frequencies are
shifted up or down, depending on the sign of the nonlinearity. If n 2 is positive, then
the nonlinear index of refraction increases as the optical intensity increases. Since a
larger index implies a smaller group velocity, the most intense portion of the pulse
sees the most phase delay because of positive Kerr nonlinearity. In this case, Eq.
(2.13) states that the "red" (longer wavelengths) portion of the pulse is shifted to the
front while the "blue" (shorter wavelengths) portion is shifted to the back.
Other Kerr nonlinearities (third-order nonlinear processes) include cross-phase
modulation (XPM) and four-wave mixing (FWM). XPM, present in multi-channel
systems, involves phase modulation of a given pulsetrain channel by the intensity
of an adjacent or orthogonal pulsetrain channel. FWM involves the interaction and
energy exchange of pulses at two wavelengths that produce Stokes and anti-Stokes
components at adjacent wavelengths if the phase conditions are matched. SPM and
XPM are special cases of FWM.
18
2.1.4
Attenuation
No material is truly transparent and fiber is no exception. As optical pulses propagate through fiber, the intensity is attenuated due to material absorption and Rayleigh
scattering. Silica in fiber absorbs in both the ultraviolent region and the far infrared
region beyond 2 pm (hence, fiber loss is wavelength-dependent).
Small amounts of
impurities lead to absorption within a wavelength window, such as hydroxide ion
implantation around 1400 nm during fabrication of single-mode fiber. Rayleigh scattering, intrinsic to all materials, arises when atomic dielectric fluctuations in the
index of refraction scatter light in all directions. This scattering dominates at short
wavelengths and sets the ultimate limit on fiber loss. Bending and splicing are also
significant external contributors to fiber loss.
The attenuation constant a is typically used to describe fiber loss. If an optical
pulse is launched with power P into a fiber of length L, then the transmitted power
at the output is
Pt = Po exp(-aL)
.
(2.14)
The parameter a is conventionally expressed in units of dB/km and SMF commercially used today typically provides an attenuation constant of 0.2 dB/km at 1550
nm.
2.2
Wave Propagation in a Nonlinear Medium
In order to analyze nonlinear wave propagation, we start with Maxwell's equations.
Consider Faraday and Ampere's Laws:
V xE=
V x H =
B9D
at
19
(2.15)
at
,
(2.16)
where E is the electric field, H is the magnetic field, D is the electric flux, B is
the magnetic flux, and J is the current density. Assuming an isotropic and uniformly
magnetic medium with nonlinear polarization PNL, we define as constitutive relations
D=cE+PNL
and
B=pH
,
(2.17)
where E and /u are the permittivity and permeability, respectively. Note that E is the
dielectric constant specific to the medium while y in this case is the free-space value,
which can be denoted as po. Taking the curl of Faraday's Law and using J
=
c-E
(Ohm's Law) yield
V x V x E+
ou
EE
_2E
+tPoEt2
- -
"
OPNL
t2
a
(2.18)
Recalling that V x V x E = V - (V -E) - V 2 E and assuming V -E = 0 1 (no charge
density in Gauss' Law, which is the case for optical fiber), we obtain
(
a2 )E =
a
__E
/a
a 2PNL
t2
(2.19)
From Eq. (2.19), we see that the nonlinear polarization acts as a source term to
the classical Helmholtz equation. Considering the waves to be time-harmonic, i.e.
a/at
-
-iw, we write
E(z, w, t)
=
8 E(z, t)ei(kz-wt)
(2.20)
and
PNL (Z,
w, t)
PNL(Z,
t) ei(kpz-wt)
(2.21)
where 6 and P are unit vectors and k and kp are the wave numbers in the direction of
the electric field E(z, t) and nonlinear polarization PNL (z, t), respectively. Substitu'This is only an approximation for an anisotropic medium.
20
tion of these field definitions into the above wave equation yields
{02
O2
+ 2ik
-
k 2)
/[t0,
- E(z, t) ei(kz-wt)
1982
2iw)
-
22wt
2a
(at2
pOE
-
-
-
W2)
.
- 2iw a
at
PPNL(Z, t)ei(kpz-wt)
We can simplify Eq. (2.22): using the dispersion relation k 2
= 1u0 Cw2
.(2.22)
and applying
the slowly varying envelope approximation. This approximation states that many
optical cycles are contained under the pulse envelope, that is,
k
OE
Oz
OE
a 2E
>
a2 E
at2
tat
(2.24)
> 2 PNL
OPNL
at
W2 PNL
(2.23)
az2
at
2
(2.25)
Eq. (2.22) now simplifies to
(az
2 k+ POE--
E(z, t)
2
=
(
-
)PNL (Z, t)ei(kp-k)z
(2.26)
Recall that w/k = c/n where n is the index of refraction so that poE = (n/c) 2 and
Eq. (2.26) becomes
nEz____
OE(z,t) poo-c
E(z, t) + n E(zt)
+
Ot
c
2n
Oz
2i(oWC(
- i)PNL (Z, t)e(kp k)z
2n
(2.27)
If we consider only the steady state and assume conductivity to be zero, which is
realistic in optical fiber, the wave equation simplifies to
OE(z, t)
Oz
SZOWC
')PNL(Z,
t)ei(kp-k)z
2n
Some general remarks:
1.) (kp - k) describes the difference in phase between the electric field
and the nonlinear polarization.
21
(2.28)
2.) If OE/Oz is real and PNL is imaginary, then there is growth or decay
of the electric field.
3.) If OE/&z is imaginary (and PNL is real) and proportional to E, then
the real part of the linear susceptibility is modulated and this induces
a change in the phase velocity.
2.3
Discussion on Regular Solitons
The term soliton was used by Zabusky and Kruskal in 1964 when they described
the particle-like behavior of numerical solutions solving the Korteweg deVries (KdV)
equation [8]. These soliton solutions remain unchanged from collisions and interactions with one another and regain their asymptotic shapes, magnitude, and speeds.
Hasegawa and Tappert theoretically showed in 1973 that an optical pulse in a dielectric fiber creates an envelope soliton and Mollenauer experimentaly demonstrated
this phenomenon in 1980 [8]. These findings are perceived to be a breakthrough in
optical fiber communications because short pulses distort after long-distance propagation due to dispersion in fiber. Proper balancing of nonlinearity and dispersion
makes soliton pulse propagation possible over long distances in optical fiber.
This section starts with the derivation of the Nonlinear Schr6dinger Equation
(NLSE), considered to be one of the most straightforward methods to model soliton
behavior. The properties of solitons described by this equation will be presented and
a discussion on implementation issues with solitons in fiber communication systems
concludes the chapter. Note that we consider here only the case where the fiber has a
uniform anomalous dispersion and these optical pulses, therefore, are generally called
regularsolitons.
2.3.1
The Nonlinear Schrodinger Equation
We start with the slowly varying envelope wave equation in Eq. (2.28) describing
propagation in a material with a nonlinear index of refraction and assume that the
22
electric field and nonlinear polarization vectors are aligned such that
OE
.
Z =Z2n
NLOWC
i(kp -k)z
Recall that the dominant nonlinear polarization term in optical fiber with a single
wavelength channel is the third-order self-phase modulation (SPM), i.e.
PNL = CoX(3)|E|2 E
CoX( 3 )EE*E
(2.30)
,
which implies that
kp = k - k-+ k
-+
k - k =0
phase-matched
-
.
(2.31)
Thus, SPM is an inherently phase-matched process. By substituting Eq. (2.30) into
Eq. (2.29), the wave equation becomes
aE + n OE
w-()1
=2 -X
2cn
C at
az
(2.32)
.
This is known as the nonlinear wave equation where the nonlinear term induces in the
propagating solution a phase shift eicO that is proportional to the E-field intensity,
i.e. q ~ X( 3)E 12 , as expected from SPM.
To model the propagation of the pulse envelope in optical fiber, we incorporate
group velocity and dispersion into Eq. (2.32).
For simplicity, we treat the Kerr
nonlinearity independently from group-velocity dispersion and consider only the linear
effects. If permittivity E(w) has a frequency dependence, so does the wave number
k(w) due to the dispersion relation. Following Eq. (2.1), we can approximate the
wave number as a Taylor expansion
k(w) ~ k(wo) +
Ok1
Ow W
(W- wo)+
102 k
2
(W- O)
.
(2.33)
2awo
To calculate the electric field spectral content in the frequency domain, we take
23
its Fourier transform
E(z, t)e(k(wo)z-wot)
-+
E(z, w - wo)ei(k(wo)z)
(2.34)
where we used the Fourier transform property FT{x(t)e(wot)} <
X(w - wo). For
small Az, the envelope must evolve as
Az
OF
= (ik(w) - ik(wo))EAz
Oz
(2.35)
,
where k(w) is the actual phase and k(wo) is the assumed phase. This effectively
corrects the envelope to consider the frequency dependence of k(w). By substituting
Eq. (2.33) into Eq. (2.35), we obtain
aE =
Ak1
(W - WO) +
92 k
(W _ wo)2
E
.
(2.36)
Recall from Fourier transform theory that multiplication by -i(W - wo) in the frequency domain translates to a temporal derivative in the time domain. Using this
property, Eq. (2.36) becomes
Define 1/v 9,
OE
Ok
Z
Ow
OE
1
-t 2
WO
Ok(wo)/Ow and k" = O2 k(wo)/0w
O2 E
2k
(2.37)
.
2
and add the nonlinear term from
Eq. (2.32) in Eq. (2.37) to obtain the nonlinear wave equation with group velocity,
dispersion, and nonlinearity:
OF + I OF= -k" 2 E ±-WX 3)lE1 2 E
Oz
V9 Ot
2 Ot 2
2cn
Note that k is generally assumed to be equivalent to
#(w)
2.
#
.
(2.38)
by convention, that is, k(w)
The latter notation will be used throughout the rest of this dissertation. It
2
This is only an approximation since f is the longitudinal component of the wave vector k (we
are considering only unidirectional propagation) and the transversal component is very small for the
fundamental mode in a dielectric waveguide such as optical fiber.
24
is possible to eliminate the first-order time derivative in Eq. (2.38) using a change
of variables. To represent the time parameter in the frame of the group velocity, we
rewrite the time variable as t -+
.u(z,
2
)z = t - /3#z and Eq. (2.38) becomes
t - (9'__
/3" 02u(z, t)
2 at 2
t)
z
-
yu(z,t)|2 u(zt)
(2.39)
,
where u(z, t) is the electric field envelope (replacing E(z, t) so that it will not generally
be confused as the electric field),
#3g
= 32 is the group velocity dispersion, and
7 = ( - n 2 )/(c - Aeff) is the Kerr nonlinearity coefficient with Aeff as the effective
core area given by
f f If (x, y) 2 dxdy
f f jf(x, y) 4 dxdy
'
which is obtained by averaging the phase shift using the modal distribution profile
f(x, y) over the fiber (integrating over tranverse directions x and y). Eq. (2.39) is the
canonical form for the Nonlinear Schr6dinger Equation (NLSE), which is an integrable
nonlinear partial differential equation frequently used to model optical soliton pulses.
Note that this equation is adequate for pulses of width To > ips such that WOTO > 1.
Also, fiber loss quantified with the parameter a can be incorporated into the canonical
NLSE. To model pulses as short as ~ 50 fs, however, a more generalized nonlinear
Schr6dinger equation should be used [9]
OnU
az
a
-u
2
/3 1 3 U
.#2 a 2 u
-5
+2i-
2
t2
6
t3
+ ... [higher-order dispersion terms]
SZ uU2
+ Y2
a(-uI 2 u) -TRURU
1 WOa
t
(2.41)
where a accounts for the intensity attenuation in fiber and TR (estimated to be
~ 5 fs) is related to the slope of the Raman gain. Note that in addition to higherorder dispersion (TOD, PMD, etc.), Eq. (2.41) also includes higher-order nonlinear
effects such as stimulated inelastic scattering (Raman and Brillouin gain) in addition
to Kerr nonlinearites (SPM, XPM, FWM).
25
2.3.2
Properties of Regular Solitons
Analytic solutions have been found for the NLSE via the inverse scattering method
[10], which maps the solution of the nonlinear PDE to solutions of linear differential
equations solvable by standard methods. If
/2
< 0 (i.e. anomalous GVD) in Eq.
(2.39), the lowest order (N = 1) solitary wave solution, or the fundamental soliton,
has a hyberbolic secant profile [11]:
U(z, t) = Ao sech
t -
#2AWOZ
exp (-iAwot)
)
x exp
[i
+
(
12Aw2)
z] exp (iq)
,
(2.42)
with the constraint that the parameters T and AO obey
-IAoI
1
r2
In Eq. (2.42), AO is the amplitude,
T
2
(2.43)
02
is the pulse width, 4 is the phase, and AwO is the
detuning from the nominal carrier frequency wo. If there is no detuning, the profile is
a simple sech function of amplitude AO with an accumulated phase delay of -YIAo I2z/2
due to the Kerr effect from the average intensity. If there is detuning by AwO, the
propagation constant changes by
#2Aw/2
in the phase factor and the inverse group
velocity changes by 32 Awo, which produces a timing shift in the argument of the
hyperbolic secant. Soliton solutions of the NLSE obey the area theorem [11] where
the area of the amplitude is fixed, i.e.
Area =
ju(z, t)Idt = r
.
(2.44)
Hence, the energy of the soliton is inversely proportional to the pulse width T. The
area theorem stipulates the amount of energy required for soliton propagation.
Higher order solitons solutions (N > 1 E integers) also exist for the NLSE. They
are obtained if the pulse width is kept fixed and the amplitude takes on values that are
integer multiples of the fundamental soliton amplitude as defined in Eq. (2.43) [12].
26
Such pulses exhibit more complicated oscillatory behavior that involves pulse breathing (compression/decompression) and pulse splitting before returning to the inital
higher order soliton waveform. Because of the inherent difficulty in implementing
N-solitons, only the fundamental (N
=
1) soliton is considered in this paper.
In the canonical form of the NLSE as shown in Eq. (2.39), an obvious interplay
exists between dispersion (denoted by
#2)
and nonlinearity (indicated by 7) in the
evolution of the pulse envelope u(z, t) as it propagates along distance z. The nonlinearity compensates for the dispersion and creates its own potential well [11]. Recall
that Eq. (2.42) is only a solution to the NLSE when the GVD in fiber is anomalous
(/32 < 0) and the blue light (shorter wavelengths) travels faster than the red light
(longer wavelengths).
This anomalous dispersion counter-balances the Kerr effect
that shifts the red light forward and pushes the blue light backward. Fig. (2-1) shows
an illustration of the compensation between dispersion and nonlinearity for soliton
propagation in optical fiber. Note that while the shape of the optical pulse envelope
is maintained due to the balance of anomalous GVD and SPM, it is not exactly the
same pulse because of a resultant nonlinear phase shift induced by the nonlinearity.
If the GVD is normal (/2 > 0), severe temporal broadening results because both
phenomena reinforce each other.
BLUE
RED
BLUE
RED
GVD
t
Iu(z,t)I
Aw
SPM
=d
t
BLUE
RED
Figure 2-1: Diagram of anomalous group velocity dispersion and Kerr nonlinearity
(self-phase modulation). Proper balancing between these two effects induces pulse
shape stabilization for soliton propagation.
27
Because of the counter-balancing effect between anomalous dispersion and Kerr
nonlinearity, an optical soliton remains stable even for long propagation distances. If
an arbitrary (non-pathological) pulse is launched into the fiber, it eventually evolves
into a steady-state soliton pulse by shedding a continuum of dispersive waves in its
transient stages. The term soliton inherits its name because of its particle-like behavior. Strictly speaking, a solitary wave, which is a solution to a class of mathematical
equations to which the NLSE belongs, is only a soliton if it emerges unscathed from a
collision or pulse-to-pulse interaction [11, 13]. While two solitons colliding into each
other experience some timing and phase shifts, they both fully recover their pulse
shape and energy. Surviving adjacent pulse collisions is desirable in multi-channel
optical systems.
2.3.3
Limiting Factors for Regular Soliton Optical Networks
Regular solitons, so-called because they require constant dispersion along the entire
fiber transmission line, seem like the ideal choice for reliable ultrafast, broadband,
and long-haul propagation. They have implementation limitations, however, that
prevent them from being the silver bullet of optical fiber communications.
These
issues include careful fiber plant physical layout, pulse distortions more pronounced
in regular soliton systems, and undesirable high power requirements.
One possible drawback of regular solitons is that they require constant (anomalous) dispersion. As a result, physical implementation issues exist. The creation of
a long link of fiber that has uniform dispersion is impractical for multi-channel systems because each wavelength sees a different GVD and it is not possible to maintain
constant dispersion over a wide range of wavelengths. In addition, fiber cables previously installed underground already do not contain anomalous dispersive fiber with
constant GVD.
A more serious obstacle to using solitons in optical fiber networks is the impairment caused by spontaneous noise in erbium-doped fiber amplifiers (EDFA). Since
fiber is intrinsically lossy, long-distance pulse propagation requires signal amplification via some gain medium, such as an EDFA, in order to compensate for the fiber
28
loss. These optical amplifiers, however, introduce amplified spontaneous emission
(ASE) noise. This effect degrades the signal-to-noise ratio (SNR) at the receiver
and causes random jitter in pulse arrival times. This timing jitter is known as the
Gordon-Haus effect [14]. At each optical amplifier in a transmission fiber system,
ASE adds a certain number of photons per mode of white noise to the signal. This
leads a small random change to the central frequency of each pulse. Because of group
velocity dispersion, the random walk experienced by the carrier frequency results in
a pulse-position perturbation, i.e. timing jitter. The variance of the timing changes
(note that the mean is zero assuming the effect is modeled as a white-noise stochastic
process) is expressed as [4, 14]
(At2)
IA
1.76hwOynsp
9
2
|(G - 1)z 3 ESO1
TFWHMLa
E
(2.45)
with h as Planck's constant, wo as the carrier frequency, n, as the ASE coefficient,
G as the gain, La is the amplifier spacing, TFWHM is the pulse width at full-width
half-maximum, and E,01 is the soliton energy. Notice that the timing fluctuations
increase with distance cubed and so the Gordon-Haus effect typically dominates after
long-haul propagation. For high data rate communication systems, randomly displaced pulses can end up in neighboring slots and thus cause detection errors at the
receiver. In Eq. (2.45), timing jitter is proportional to GVD. So one way of reducing
timing jitter is to decrease dispersion. While Eq. (2.45) predicts that zero dispersion
completely eliminates jitter, solitons require a finite amount of dispersion for them to
exist because the soliton energy is proportional to the dispersion from Eqs. (2.43) and
(2.44). If the dispersion is very small, then the soliton has very small energy, which
compromises the SNR. There has been much research on reducing timing jitter via
other means such as narrow-band filters (passive methods) and in-line synchronous
modulators (active methods). Methods involving the sliding guiding filter principle
[4, 15], however, are undesirable to system engineers because this requires different
filters in subsequent amplifier pods to slowly guide the pulse spectrum away from the
ASE noise.
29
Another major implementation issue involved with regular solitons is the requirement for high optical power levels. This is primarily due to the fabrication constraint
where fibers with uniformly low dispersion is very difficult to manufactured. In order
for the SPM to balance GVD, the intensity of the optical field must be high enough
to cause a change in the index of refraction. Typically, the power and energy levels of
regular solitons are quite high in order to induce the required SPM magnitude since
the nonlinear index coefficient n 2 is relatively weak. Such high power requirements
are inefficient, impractical, and often unacceptable in communication networks.
As an aside, soliton systems have yet to win a competitive edge over linear techniques currently employed in the market.
These "linear" techniques counter the
philosophy of soliton propagation by suppressing or minimizing the Kerr nonlinearity
instead of exploiting it. One example is nonreturn-to-zero (NRZ) pulse transmission.
This modulation format consists of rectangular ones (pulse train of bits) forming a
continuous block if two or more ones ("1") occur consecutively and falling to null if a
zero ("0") occurs. Fig. (2-2) provides a comparison between NRZ and soliton pulses.
At today's 10 Gbit/s data rates, these linear techniques work remarkably well and
Data
1
1
1
0 0 1
0
1
1
0
NRZ
Soliton
Figure 2-2: Comparison between NRZ and soliton transmission formats for the given
data stream.
have been implemented commercially. They also have power/energy levels far below
those required for regular soliton communications. For tomorrow's higher 40+ Gbit/s
data, however, soliton-based optical systems may be required to manage higher order
effects such as PMD and to provide spectral efficiency.
30
Chapter 3
Background on
Dispersion-Managed Solitons
This chapter describes a new class of optical solitary waves called dispersion-managed
(DM) solitons. The dispersion map, which is the basic structure in creating such
pulses, is discussed, along with the dynamics and evolution of DM solitons. Some of
the underlying physics is revealed via numerical simulations and variational methods.
Finally, properties of DM solitons are presented in comparison to regular solitons in
order to provide some motivation for this thesis.
3.1
Introduction to Dispersion-Managed Solitons
A new class of solitons called dispersion-managed solitons can be created using dispersion management [16]. Dispersion management is a technique that uses varying dispersions, rather than uniform dispersion, in a link of fiber. Dispersion compensation
occurs when one fiber segment of a particular dispersion is used to counter-balance
the pulse broadening in another segment of opposite dispersion. A DM soliton is
manifested on a fiber with a dispersion map, which consists periodic segments of dispersion of alternating signs. Usually, these periodic segment structures, or unit cells,
are placed one after another to produce periodic dispersion compensation in the fiber
link. An example of a dispersion map is depicted in Fig. (3-1).
31
Unit Cell
Norm
t>
Anom
_
rve
Figure 3-1: Example of a symmetric two-stage dispersion map with a path-average
anomalous dispersion.
The simplest dispersion map is constructed from two fiber segments, one anomalous
and the other normal. The lengths of the segments can differ. The path-average (net)
dispersion associated with a particular map is given by [16]
-
ave
_ #"L, +/ La
- L, + La
(3 1)
'
#" and Lj refer to the jth segment's dispersion and length, respectively. For consistency throughout this section, parameters with the subscript 'n' correspond to the
normal dispersion fiber and those with the subscript 'a' correspond to the anomalous dispersion fiber. Another characteristic of a dispersion map is the map strength,
defined as
S_ | (#'Ln - 0"La)|
(3.2)
(32
FWHM
where
TFWHM
is the minimum full width at half maximum (FWHM) of the pulse when
unchirped. The map strength represents a single dimensionless quantity that measures the difference between the total dispersion accumulated in both fiber segments,
rather than the difference of the two dispersion values. Essentially, S measures the
32
spreading factor of the pulses. Higher magnitudes in either segment lead to stronger
maps since a larger dispersion swing spreads the pulses more. Pulses with shorter
pulse durations also disperse quicker. Shorter pulse widths therefore effectively lead
to higher map strengths.
Within each period of the dispersion map, the pulse undergoes breathing, which
is marked by compressing and broadening in time [16]. Unlike regular solitons, DM
solitons are not continually stationary but periodically stationary. The pulse returns
to its original pulse shape after each map period (a unit cell). A numerical simulation
of a DM soliton traveling through a unit cell of a dispersion map is shown in Fig.
(3-2). The DM soliton temporally broadens and compresses twice within each map
period and the chirp in each of the fiber segments has opposite signs. If power loss
is neglected and the nonlinear coefficient is assumed to be the same for both fiber
segments, then the pulse is narrowest, and hence unchirped, in the middle of each
fiber segment. The pulse is also broadest and most strongly chirped at the boundaries
between the anomalous and normal fiber segments.
0-W
Distance
Time
Figure 3-2: Numerical simulation of DM soliton in one unit cell of simple two-stage
dispersion map.
To achieve DM soliton propagation, an.important design parameter is to maximize
33
the coupling of energy from the launched pulse into the soliton and to minimize
the generation of dispersive radiation. In the dispersion map, the pulse experiences
periodically varying chirp during propagation through each unit cell. If the optical
source used to generate short pulses is chirped, the source must be located at a
position in the unit cell where the chirp of the source matches that in the unit cell in
order to maximize DM soliton coupling efficiency [16]. If the source is assumed to be
transform-limited and generating unchirped pulses, the dispersion map should begin
at the mid-point of one of the fiber segments in the unit cell where steady-state pulses
are chirp-free. This argument provides the reasoning for implementing the first unit
cell in a dispersion map with half the length of the anomalous (or normal) dispersion
segment used in subsequent two-stage unit cells in the dispersion map. If this map
starts with the full length rather than half-length and the launched pulse is unchirped,
then there is a large transient response that sheds energy into dispersive waves until
the pulse reaches the steady-state where it is chirp-free at the center of subsequent
fiber segments in the map. Ideally, a pulse can be launched anywhere on the dispersion
map period, provided that the chirp of the source is appropriately chosen for the
launch location in the map. If there is gain (or attenuation) in the dispersion map,
then the chirp-free point is not in the center of either segment anymore. In this case,
the chirp of the source can be adjusted in order to move the unchirped location to
the middle of the anomalous and normal fiber sections in a two-stage dispersion map.
A feature that differentiates a DM soliton from an ordinary soliton is that solitary
wave propagation in dispersion maps does not assume stable pulse shapes that are
hyperbolic secant [16]. As map strength S becomes stronger, the shape changes from
a sech-profile to a Gaussian. The time-bandwidth product increases from 0.32 (sech)
to 0.44 (Gaussian). When the map becomes even stronger, pulses may assume shapes
having even higher time-bandwidth products [16].
34
3.2
Methods for Theoretical Analysis
Numerous theoretical studies have been done on dispersion-managed solitons in order
to shed some physical insight on such optical pulses. While numerical simulations of
pulse propagation determined by the NLSE yield accurate physical predictions, the
solutions are not analytic since the addition of dispersion management renders the
NLSE non-integrable and the trial-and-error process of selecting parameters is tedious.
Although such techniques do not rigorously describe the details of pulse propagation,
approximation methods can be used to generate analytic solutions, which are typically
a set of coupled ordinary differential equations. The two most frequently employed
methods for soliton numerical modeling are the split-step Fourier method (rigorous
numerical simulation) [9] and the variational approach (approximate analytic technique) [12] and each of these is discussed in the following sections. Note that these
algorithms were initially applied to regular soliton transmission and then extended
to describe the propagation of DM solitons.
3.2.1
Numerical Simulation: The Split-Step Fourier Method
The most popular and perhaps most efficient numerical algorithm in solving the
NLSE is the split-step Fourier method [9].
This technique uses the Fast Fourier
Transform (FFT) to propagate the waveform through dispersive fiber in the absence
of nonlinearity and treats the nonlinearity as a lumped element between the steps.
Since the split-step algorithm uses the FFT (which scales by N log N, where N is the
number of operations), the relative speed of this method is approximately one order
of magnitude faster than finite-difference methods in achieving comparable accuracy
[9]. The computational efficiency of the split-step Fourier method is thus one of the
reasons for its popular usage in simulating pulse propagation in nonlinear dispersive
medium.
To implement this algorithm, it is useful to write the NLSE expressed in Eq.
35
(2.39) in the form
(3.3)
u=(D+N)u
where D is the differential operator accounting for dispersion (and also absorption)
in a linear medium and N is the nonlinear operator accounting for the effect of fiber
nonlinearities. These operators are defined as
#'/2 &2 1a\
D = -_ a2 at
2
N =_Z'yn2
(3.4)
'
.
(3.5)
Note that the term inside the parentheses in the linear operator b accounts for the
loss. While the dispersive and nonlinear effects typically act on an optical pulse simultaneously during propagation through fiber, the split-step Fourier method generates
an approximate solution by propagating a small distance step size f and assuming
that the linear and nonlinear effects operate independently. In the simplest case, the
propagation from z to z + t can be executed in two steps, the first where dispersion acts alone (N = 0) and the second where nonlinearity acts without dispersion
(D = 0). The mathematical solution to Eq. (3.3) is [9]
u(z + f, t) ~ exp(Ne) exp(Th)u(z, t)
(3.6)
.
The dispersive effect can be solved within the Fourier domain (since dispersion is
linear). If we take the Fourier transform of the linear part of Eq. (3.3), i.e. consider
only the
b operator,
(9u
az
we have
#2
- Z=t2
2 t
2
->
2
36
U2=
dz
Z-
2
~U./3T
a~
2,U
U
2
(3.7)
where U(z, w) denotes the Fourier transform of u(z, t). The solution to Eq. (3.7) is
U = Uoex
()p
(-2
where
Uo
2
U)
(3.8)
,
is determined from initial or boundary conditions on z and w. In order to
find the optical field in the time domain, we simply take the inverse Fourier transform
of the field in the frequency domain, i.e. u(z, t) = Y-0{U(z, w)}. The nonlinear part
of Eq. (3.3) is then solved in the time domain. The solution using only the N operator
is
UNL = Uo exp (Y IUO12 Z)
(3.9)
,
where Uo can be found from initial or boundary conditions. Note that this solution
for the nonlinear contribution neglects Raman (and Brillouin) gain. If these nonlinear
effects are included, it may be more convenient to express the optical field in terms
of polar coordinates and solve for the resulting optical field in magnitude and phase
components. The accuracy of the split-step Fourier method (using the simple twostep case) can be estimated by expressing an exact solution to Eq. (3.3) in terms of
the operators themselves and using the Baker-Hausdorff formula for noncommuting
operators [9] (the split-step here ignores the noncommuting nature of b and N). It
is found that the algorithm described above is accurate to second order in step size f.
It is possible to improve the accuracy over the simple two-step case by accounting
for the nonlinearity in the middle of the dispersion segment rather than as a boundary
segment. That is, we write the solution to Eq. (3.3) as
u(z + f, t) ~
exp (-b)
exp
(7
N(z')dz'
exp
fIb) u(z, t)
.
(3.10)
Essentially, we treat the nonlinearity as a lumped element in the middle of the step
size F as the optical field propagates along a linear medium. Because of this symmetric
form of the operators, this approach is known as the symmetrized split-step Fourier
method [9]. If the step size f is small enough, the middle exponential in Eq. (3.10)
37
can be approximated as exp(Nl). The improved accuracy for this approximation is
that the dominant error term is reduced to third order in step size f. The accuracy is
further improved by evaluating the integral in Eq. (3.10) using the trapezoidal rule
and approximating the integral as [91
N(z')dz'
2[N(z) + N(z + f)]
(3.11)
Implementing Eq. (3.11) is not simple, however, because N(z+f) is not known yet at
the location z + f/2 due to causality. An iterative procedure is needed where N(z + f)
is replaced by N(z), u(z + f, t) is estimated via Eq. (3.10), and this estimated field
is used to calculate the new value of N(z + f). Although this additional iteration
may sound time-consuming, the overall computation time is actually reduced if the
step size can be increased due to improved accuracy of this symmetrical numerical
approach.
Although implementation of the split-step Fourier transform is straightforward by
following the above recipe of steps, some caveats exist when using this algorithm.
The step sizes in space and time must be chosen appropriately in order to maintain
accuracy and the appropriate step sizes can vary depending on the complexity of the
simulated problem. Typically, trial and error runs can determine optimum values for
computational speed and accuracy. Another concern is to ensure the time window is
sufficiently wide so that the pulse energy remains inside the window at all times. This
is an issue especially for high map strengths where pulses spread in time very rapidly
and run the risk of exceeding the window boundaries. Since the FFT involves periodic
boundary conditions, energy exceeding one edge of the time window re-enters from
the other edge, which can lead to numerical instabilities. Nevertheless, the speed
advantage over other rigorous numerical algorithms such as finite-difference methods
makes the split-step Fourier method a powerful tool to analyze pulse propagation
through fiber.
38
3.2.2
Approximate Method: The Variational Approach
In some applications, a rigorous numerical solution to the NLSE is not required. In
these cases, approximate methods are used to obtain computationally efficient analytic solutions that provide physical insight. The variational approach [12] is an
example of approximate methods which yield reasonable solutions to the NLSE. This
technique can be used to analyze various aspects of DM solitons including average dispersion, power, and map strength. The variational approach involves trial functions
used to describe the main characteristics of pulse evolution as dictated by the NLSE.
The approach provides useful explicit analytical expressions for the evolution of essential soliton characteristics such as pulse compression/decompression factor (related
to pulse width), the maximum pulse amplitude, and the induced frequency chirp [12].
One source of error using the variational method results because the shape of the trial
function is preserved throughout the simulation, which does not account for changes
in pulse shapes. Additionally, significant higher-order soliton effects (such as pulse
splitting) cannot be directly analyzed through this approach and are approximated
instead as pulse broadening. Nevertheless, if we are interested only in the slow pulse
dynamics in soliton propagation, the variational approach provides a description of
the complicated interplay between dispersive and nonlinear effects [12].
The following discussion of the variational method is presented in the context of
regular solitons, as originally done in [12]. The approach is applied to the dispersionmanaged case later in this section. The approximate analysis essentially involves a
Ritz optimization procedure based on the variational functional corresponding to the
NLSE. The NSLE can be restated as a variational problem in terms of the Lagrangian
[12]
i(Ou*
L = - (
2
U
az
&Bu\/
- u*aU
+
19z
39
2
2
B
aUa
at
2
-2
I1
.
(3.12)
Using calculus of variations, the NLSE is obtained via the variational principle
,u*, au, a*, a
J Z *z atat
dzdt = 0
(3.13)
,
where 6 denotes the first variation of L. Eq. (3.13) must obey the Euler-Lagrange
conditions where the first variation of the Lagrangian disappears with respect to one
of its parameters, i.e.
6L
6u
=0
-+=0
aL
19U
0
aL
a aL
at a g
az ()a
.(3.14)
Note that the Euler-Lagrange conditions are in the form for a functional with two
independent variables, as is the case in Eq. (3.14) for L with z and t.
According to the Ritz procedure [12], the first variational of the functional is made
to vanish given a set of suitably chosen trial functions. A Gaussian ansatz is typically
used since the first term in a Gaussian-Hermite expansion accurately describes DM
solitons [17]. In addition, the Gaussian pulse shape reproduces exact solutions to the
NLSE within the linear limit of the variational equations. Hence we specify the pulse
evolution or trial solution as a Gaussian in the form
u(z, t) = A(z) exp
(-
t()
+
ib(z)t2
(3.15)
,
where A(z) is the complex amplitude, a(z) is the pulse duration, and b(z) is the
frequency chirp parameter.
Note that the pulse parameters can vary with propa-
gation distance and the complex amplitude can be separated into its real part (the
magnitude) and its imaginary part (the phase), i.e. A(z) =
IA(z)I
exp(io(z)).
Inserting the trial function u(z, t) given by Eq. (3.15) into the variational principle
expressed in Eq. (3.13), we obtain the reduced variational problem [12]
6J(L)dz=0 ,
40
(3.16)
where
(L) = f
ia A dA* - A*dA) + A2a
LGdt
-
+ v/2 -yaIA|4j
b2 +
a3|A12
(3.17)
and LG in Eq. (3.2.2) is the result of evaluating the Lagrangian L with the Gaussian
ansatz u(z, t) in Eq. (3.15). The reduced variational principle as expressed in Eq.
(3.16) generates a set of ordinary differential equations for the pulse parameters A,
a, and b, which together determine pulse evolution.
Using the Euler-Lagrange equations, similar to that in Eq. (3.14), we obtain the
following variational equations [12]:
db
d A*
d
(-iaA*) = -i'a dz + A*as
dz ±
dz
dz
0
6(L)
6AL) = 0 6A
- a A*
2
6(L)
6A*
d
dA
-(iaA)=-iaz-+
dz
dz
-A*=
3aA
-2
/
-
i
-)+227ya|A12A
a4
,(3.18)
3 db
Aaz dz
-2
6(L)
6a = 0
4b2+
(.8
+2'IIIA
4b2
a
+ 2xyaIA| 2 A
+ a)+2/-aA2
,
(3.19)
+ V-yIA 14 = 0
2 a2
,
(3.20)
12a2db
dA\
dA*
A dz -A* dz + 3A2a2
- 3/ 2 a2 b2 +
6(L) = 0 -
d (a3 |A 12 ) = -4
-
2 ba
3
A12
.
(3.21)
When Eqs. (3.18) and (3.19) are multiplied by A and A* respectively, and the results
41
are subtracted and added, we obtain the following [12]
-- (a|A 2) = 0
dz
i
A*
- A dz
A12
a2
a2
(3.22)
,
4b2 +
+ 2V2'YIA|12
.
(3.23)
Eq. (3.22) implies a constant of motion, that is,
a(z)A(z)12 = const = aoIAo12 = EO,
(3.24)
which indicates that the energy of the pulse stays the same. Note that because of this
conservation of energy, the variational approach in [12] assumes a lossless medium.
Since a(z)A(z)12 is constant, Eq. (3.21) becomes
da
-
-2#
2 ab
.
(3.25)
dz
By comparing Eqs. (3.20) and (3.23), we obtain
db
/3'A
aT - 2# 2 ab2 + 2
dz
20
2
|A3.26
=o
0
-,F a
,(3.26)
which can be combined with the derivative form of Eq. (3.25) to result in
-23
2
A2l
(3.27)
The second-order differential equation in Eq. (3.27) yields an equation for the variation of pulse width a(z), which in turn solves for the chirp parameter b(z) using Eq.
(3.25) [12]. The magnitude of the complex amplitude, IA(z)l, is determined by the
constant of motion in Eq. (3.24) and the phase from the complex amplitude, '(z), is
determined using Eqs. (3.23) and (3.26) [12].
The equations for the pulse width, chirp, magnitude, and phase of the pulse solve
the variational problem [12] and these parameters provide a sufficient description
42
for the evolution and behavior of the soliton. While the above discussion on the
variational approach pertains to regular solitons, the same method can be applied to
DM solitons using the same form for the set of coupled ODEs. One can infer the
average dispersion and peak power from the variational parameter quantities. The
key difference between regular and DM solitons is the dispersion map. In order to
analyze DM soliton propagation using the variational method described previously
for regular solitons, we simply substitute the DM soliton average dispersion defined
in Eq. (3.1) for the uniform
#2 dispersion
parameter used for regular solitons.
Several examples utilizing the variational approach in the context of DM solitons
are shown in references [4, 2, 18, 19, 20, 21]. In these cases, the pulse duration and frequency chirp during pulse evolution are monitored by solving Eqs. (3.25) and (3.26).
A periodic (two-stage) dispersion map is modeled with symmetric and antisymmetric
boundary conditions where the pulse width is the same at the beginning and end of
the map and the chirp has opposite signs, i.e.
a(La) = a(La + L) =ao
(3.28)
b(La) = -b(La + Ln) = bo.
(3.29)
The dispersion can either be specified for each segment of the map (if the evolution
equations for pulse width and chirp are used to derive eigenvalue equations with DM
soliton parameters [22, 18]) or found implicitly from the calculated pulse width and
chirp (if those evolution equations are written as a single second-order differential
equation in the form of Eq. (3.27) [2]). The dependence on peak power, energy,
and map strength for DM solitons is determined from the dispersion parameters and
closed-form expressions derived from the variational equations. The fiber medium is
assumed to be lossless because of the required invariance of energy in the variational
method from Eq. (3.24).
In general, the variational approach [12] is utilized on the nonlinear Schr6dinger
equation to obtain a reduced parameter space involving power, pulse duration (i.e.
pulse width), chirp, and phase. This mathematically yields a set of coupled ordinary
43
differential equations (Euler-Lagrange equations), which are solved to generate plot
relations among power, net dispersion, and map strength. While it is an approximate
technique, the variational method nevertheless is a useful means of analyzing pulse
behavior and evolution.
3.3
Behavior and Characteristics of DM Solitons
One remarkable result that is easily observable from using the variational approach
is that DM solitons can propagate in dispersion-balanced (zero net dispersion) maps
and even in maps that have normal average dispersion [23]. In balanced maps, this
phenomenon is very interesting because while the pulse is nonlinear, there is no dispersion with which to balance the nonlinearity in the conventional sense [23].
A
resulting benefit from net zero dispersion is that dispersion-dependent timing jitter
can be virtually eliminated. In maps with normal average dispersion, it is surprising
for solitons in dispersion-managed systems to exist while regular ("bright") solitons
in uniform fiber, cannot exist since they require the balancing effects of anomalous
GVD and SPM. The propagation of DM solitons in the normal average dispersion
regime is possible if the map strength is above a critical level [2]. When the energy
(or power) is plotted against net dispersion at some constant pulse width (or map
strength) as shown in Fig. (3-3), two energy solutions exist for each dispersion value
in the net normal dispersion regime. Recall from Eq. (3.2) that the map strength
is increased by creating a greater dispersion imbalance between the two half-cells in
a two-stage map or by reducing the pulse width. If the map strength is stronger,
the curves penetrate deeper into the average normal dispersion regime, as shown in
Fig. (3-3). Also, for each map strength parameter, the pulse energy increases when
the transition is made from the net normal to the net anomalous dispersion regime
because of the higher energy required for solitons to propagate in uniform anomalous dispersion. If we consider the pulse solitons on the lower-energy branch of the
net normal dispersion regime in Fig. (3-3), we notice the quasi-linear characteristic
of the energy dependence near net zero dispersion. This characteristic has exciting
44
implications for fiber optic communication systems because Gordon-Haus jitter and
soliton pulse energy requirements are greatly reduced from the regular soliton pulses
discussed in Chapter 2.
. I I I-
120
100
(U 80
FWHM=11ps
16psU
60
z
18p
40
20
-0
-
L=100k
k"=17ps 2/kI
0
0.2
0.4
0.6
0.8
j
i
1
Ak" (ps 2 /km)
Figure 3-3: Variational plot of DM soliton energy versus net normal dispersion with
pulse width as a parameter (provided by Prof. Haus). The circles on the plot represent
direct numerical simulations.
One may wonder how a dispersion-managed soliton can physically exist in the
normal dispersion regime. We have shown how a regular soliton is formed with the
right balance between GVD and SPM in a fiber with uniform anomalous dispersion.
DM solitons, however, can propagate in the normal average dispersion regime with
the following reasoning. While the pulse is chirp-free in the center of either segment
(in the lossless case), the pulse width is shorter in the anomalous dispersion regime
than in the normal dispersion regime. Hence, the bandwidth of the pulse becomes
narrower during propagation in anomalous dispersion and wider in normal dispersion
[6, 18]. The bandwidths must be continuous at the boundary between the two fiber
segments. As a result, the bandwidth overall is larger in anomalous dispersion than
in normal dispersion. If we consider the Gaussian pulse defined in Eq. (3.15) and
45
take its Fourier transform, i.e. U(z,w) = B(z) exp(-(1 - ib)w2 /2Q 2 ) where B(z) is
the complex amplitude in the Fourier domain and Q is the bandwidth, we can relate
frequency chirp and bandwidth in the following way:
=
a2Q 2
1 + b2
(3.30)
.
Since linear chirp is directly proportional to the bandwidth squared, this means that
if the dispersion in the two fiber segments of a unit cell is equal in magnitude but
opposite in sign for net dispersion of zero, then the pulse essentially acquires a net
anomalous chirp [6]. This can be interpreted as effective anomalous dispersion. An
expression for this shift is found in [6]:
(3"Q 2 )
ef
=
2
(Q )
27E
3
(/")
where E refers to the pulse energy,
Qb
Qb
[1
-
(1 + 4c 2 -1 / 2 ]
(3.31)
,
is the bandwidth at either boundary between
the segments, and c is the chirp parameter assuming net zero dispersion, i.e. c =
Ca,n =
/,nLa,nQ2/4
(note that Ca
-ca).
The shift in effective dispersion explains
how DM solitons can exist when the net dispersion is normal. Solitons exist only if
the effective dispersion is anomalous, that is, if Eq. (3.31) is negative. This can be
,) is normal (positive
true even if the average dispersion ((0")
#"
/32)
provided
that the chirp parameter c can balance it. Hence, chirp is required for DM solitons
to operate in the normal dispersion regime. Propagation of DM solitons in the pathaverage normal dispersion regime thus involves the balance of chirp, dispersion, and
nonlinearity.
Overall, the change in pulse shape and existence at zero or net normal dispersion
reinforce the notion that dispersion-managed solitons are indeed a new class of optical fiber solitons [16]. As an aside, the name dispersion-managed soliton may be a
misnomer. A DM soliton may not really be a "soliton" in the original meaning of the
word, even though it returns to its original shape after each period in the dispersion
map despite temporal breathing. DM solitons with Gaussian profiles are not exact
solutions of the NLSE, although the variational approach is used to analyze DM soli46
tons. They also do not recover from pulse-to-pulse collisions, like regular solitons. A
dispersion-managed "soliton" is more appropriately a solitary wave, a more general
class of pulses of which a soliton is a member. DM solitons are also analogous to the
pulses generated by stretched-pulse lasers, which use normal and anomalous dispersion in the fiber ring for operation in the normal dispersion regime [4]. Furthermore,
DM solitons can be considered as nonlinear Bloch waves with a periodic scattering
potential and with no continuum shedding [24].
47
Chapter 4
Experimental Search for
Lower-Energy DM Solitons
The theoretical studies discussed in the previous chapter indicate that low-energy
dispersion-managed solitons exist when operating in the path-average normal dispersion regime. This is a crucial discovery since these DM solitons with lower energy
requirements offer an advantage over regular solitons that need high powers and also
exploit nonlinearity to combat dispersion, which is not accomplished by linear techniques. While numerical investigation on DM solitons is usually limited to the ideal
lossless case (especially since the variational method assumes that energy is invariant), no physical reason exists for why the lower-energy solutions cannot propagate
over low-loss single-mode fiber. The crux of this thesis seeks to demonstrate experimentally the existence of these lower-energy DM solitons. At the time of writing
this thesis, no one has claimed the experimental observation of such optical pulses,
although chirped-return-to-zero (CRZ) results presented in [25, 26] may arguably describe the low-energy DM solitons. This chapter describes the experimental setup
and measurements, the process of experimentation in locating the lower-energy DM
solitons in the net normal dispersion regime, and the data that demonstrates evidence
for the existence of these optical pulses.
48
4.1
Experimental Objective
The initial intent of this experimental endeavour is to produce periodically stationary
pulses in a recirculating fiber loop with a specifically designed dispersion map in the
path-average normal disperion regime. While it is preferably to launch the "right"
pulses into the loop in order to eliminate all transients so that the steady-state pulse
appears during the first loop period, this is rather difficult due to equipment limitations, such as difficulty in specifying chirp of the initial pulses and controlling the
pulse shape. A realistic goal then is to launch something close to the ideal parameters and have the steady-state dispersion-managed soliton emerge after the first few
periods around the loop. If the pulse is truly periodically stationary, it returns to
the same pulse shape, width, and energy at a fixed location within the loop everytime. Because optical amplifiers produce inevitable amplifier spontaneous emission
(ASE) noise that eventually deteriorates the pulses around the loop, pulses that have
achieved a periodically stationary state will not remain so for many roundtrips around
the loop if the noise is not managed well. Since the goal of this thesis is to establish the existence of DM solitons at net normal dispersion with relatively low energy
levels, the inherent noise issue will be postponed for now. Experimental observation
that pulses remain convincingly periodically stationary during some loop periods before succumbing to ASE noise will provide evidence for the thesis objective. Future
work will deal with the noise problem for long-haul propagation with amplifiers as
necessary in real optical fiber systems.
4.2
Experimental Setup
This section describes the apparatus used in setting up the recirculating fiber loop
experiments. Specifications and measurements relevant to data acquisition and interpretation are reported.
49
Laser Source
4.2.1
The transmitting source that generates optical pulses to be launched into the fiber
loop is a tunable modelocked fiber ring laser. It is capable of producing Gaussian
pulses with a width of 2-3 ps. The source is modulated at 10 GHz by a continuouswave (CW) generator such that the pulse separation is 100 ps.
To estimate the chirp in the pulses produced from the laser source, the timebandwidth product is calculated to determine whether it is approximately transformlimited. The full width at half maximum (FWHM) of the pulse duration can easily
be determined from an autocorrelation of the laser pulses and the spectral width can
be measured in an optical spectrum analyzer (OSA). These measurements for the
laser tuned at 1550 nm are shown in Fig. (4-1). If the pulse shape is assumed to be
Gaussian, the actual pulse width can be inferred from the autocorrelation width by
dividing the latter by a square root of two. For Gaussian pulses, the transform-limited
time-bandwidth product is about 0.44.
Autocorrelation of Laser Pulse
Spectrum of Laser Pulse
-10
4500
40003500
-20
-
3000-
02500-S
FWHM =
3.5 ps
-
-30 -
4
FWHM =
1.3 nm
_
2000 -
1500 -
-50 -
1000
-60
5000
-10
-5
0
Time [ps]
5
-7C
1520
10
1530
1540
1550 1560 1570
Wavelength (nmj
1580
1590
Figure 4-1: Autocorrelation (left) and optical spectrum (right) of the transmitter
laser pulse. The autocorrelation FWHM translates to At = 2.5 ps and the bandwidth
tranlates to Av = 160 GHz for a time-bandwidth product of about 0.4, which is close
to the Gaussian transform-limited state. The pedestal in the spectrum is ASE noise
produced by an optical amplifier.
50
4.2.2
Dispersion Map and Measurements
The first step in creating dispersion-managed solitons is to design an appropriate
dispersion map. This map determines the map strength given the pulse width of the
propagating optical pulses and sets the behavior and evolution of the DM solitons
throughout the fiber loop. The dispersion profile of the loop should preferably have
the zero average dispersion point at the center wavelength of the transmitter's tunable
range so that sufficient accessible wavelengths exist for net anomalous (increasing the
wavelength) and net normal (decreasing the wavelength) dispersion. The dispersion
map consists of unit cells periodically replicated such that the fiber loop defines one
unit cell. The loop contains two segments of opposite dispersions labeled "half-cells,"
a loosely used term since the anomalous and normal segments may have different
lengths.
Before specifying the fiber needed to design the dispersion map, a technique to
measure the dispersion characteristics of the fiber must be developed. This characterization is achieved by measuring the relative group delay between different wavelengths
such that the dispersion (GVD) is the derivative of that delay with respect to the
wavelength (this is the D parameter, as opposed to
/32).
A setup of the dispersion
measurement method following this principle is shown in Fig. (4-2). A tunable semiconductor laser serving as a broadband source is modulated with an electro-optic
(EO) modulator (a Mach-Zehnder interferometer). A signal generator sets the modulation rate by providing an RF signal to the EO modulator. A DC voltage source
is used to bias the EO modulator in the linear regime at quadrature. Setting this
DC bias at quadrature is done by providing a voltage such that the operating point
is on the maximum (i.e. completely switching out the signal) and then lowering that
voltage by 3 dB (half of the switching voltage V,) with the aid of an inline optical
power monitor. The modulated signal is sent through some fiber whose dispersion
is to be tested and then amplified via an erbium-doped fiber amplifier (EDFA). A
narrow-band filter is used to eliminate most of the ASE noise from the optical amplifier. An OSA can also be used to ensure that the operating wavelength lies within the
51
Tunable
~
PC
Laser
MVZM
RF
SIGNAL
PM
BIAS
Signal
FIBER
TO BE
TESTED
EDFA
BPF
PC - Polarization Controller
MZM - Mach Zehnder Modulator
PM - Power Monitor (inline)
TRIGGER
BPF - BandPass Filter
PD - PhotoDetector
PM
O
Sampling
t,
PDSpectrum
Scope
KRF AMP 9/0
Aaye
Figure 4-2: Experimental setup for dispersion measurements of fiber sections.
filter passband. The filtered and amplified optical signal is detected by a photodiode
and the electrical signal is amplified with a bandlimited RF amplifer and then passed
to a digital sampling oscilloscope. The same signal generator used to set the rate
for the EO modulator triggers the scope. The total group velocity delay relative to
this triggering signal is measured for a sequence of multiple wavelengths. Since each
wavelength experiences a different group delay due to dispersion, the measured relative delay from the triggering signal changes for each wavelength tuned in the laser
source. Hence, different shifts of the scope traces for each wavelength are observed.
The dispersion for each wavelength can be calculated by taking the derivative of the
group delay with respect to wavelength. The aggregate of measurements for a range
of wavelengths determines the dispersion profile of the tested fiber section.
Some experimental implications involved with this dispersion measurement technique must be considered. One is that the total dispersion (i.e. DL, where L is the
length of the tested fiber) is measured, not the D parameter (in units of ps/nm-km).
Therefore, if the fiber is too weakly dispersive or its length too short, any change in
the group velocity delay experienced by different wavelengths may not be perceptible
in the sampling oscilloscope. It is possible to enhance the changes of the group delay
52
relative to the triggering signal by modulating the source pulses at a higher frequency
so that the ratio of the relative shifting to the period of the modulation is larger. Care
must be taken not to set the modulation frequency too high because if too much total
dispersion is present or if the repetition rate of the modulated signal is too high, then
the shifting from the triggering signal may be too great such that the group delay may
overlap the period of the modulation, an occurrence that introduces discontinuities
in the data. Another issue of concern is the tendency of the EO modulator to drift
from the operating point on the transmission curve. Operation in the linear regime
is monitored with an inline power meter and maintained by adjustment of the DC
bias voltage as needed so that the operating point remains at quadrature. Finally, in
order to ensure consistency among the data measurements, the average power going
into the sampling oscilloscope should remain the same for all wavelengths. An inline
power monitor equipped with an attenuator can control this.
The design of the dispersion map used in this experiment consists of four dispersion
sections, which make up the anomalous (three fiber sections) and normal (one fiber
section) segments. The first anomalous fiber section consists of approximately 50 km
of AllWaveTM fiber. AllWaveTM is anomalous single-mode fiber similar to Corning
SMF-28 except the loss induced from OH- impurities during fiber manufacturing at
the 1400 nm wavelength regime is eliminated. Since the operating wavelength regime
for this experiment is around 1550 nm, AllWaveTM fiber is effectively the same as
other anomalous single-mode fibers in terms of dispersion, nonlinearity, and loss. The
second fiber section is about 25 km of AllWaveTM fiber and the third section is made
up of Corning's SMF-28. The normal dispersion segment comprises one fiber section of
about 15 km dispersion compensated fiber (DCF). DCF has a smaller core size and so
it effectively has higher nonlinearity than the other fibers used in the loop. It also has
more dispersion so a shorter span is required to compensate the anomalous dispersion
of the other three fiber sections. The length of the anomalous section consisting of
only SMF-28 is modified (with the other three fiber sections fixed) until the desired
dispersion profile for the entire loop is obtained. Ideally, we wish to achieve net zero
dispersion at the wavelength of 1550 nm. Uptuning (i.e. greater than 1550 nm) the
53
transmitting laser source provides operation in the net anomalous dispersion regime
while downtuning gives operation in the net normal dispersion regime. The dispersion
measurements for each of the four fiber sections are shown in Figs. (4-3, 4-4, 4-5, 4-6)
and the calculated dispersion profile of the loop is given in Fig. (4-7). Each figure
contains two plots. The top plot shows the measured relative group delays for each
wavelength (the tuning range is between 1530 and 1564 nm with a measurement taken
at every 2 nm). The circles indicate the measurements and the solid line represents
the best second-order polynomial fit. The bottom plot shows the derivative of the
top plot with respect to wavelength. This graph gives the dispersion for the tested
fiber spool. The dispersion swing between the anomalous and normal segments is
quite high because of the large local dispersion in the normal segment. In addition,
chirp-free pulses produced by the modelocked fiber laser can be as short as 2 ps. The
dispersion map thus have a very large map strength where there is intense spreading
of the pulse. This implies that third-order dispersion, which especially affects pulses
with a greater spectral bandwidth, may be non-negligible in these experiments.
-40000
30000
0
-
-.-.-.-
~20000-
> 10000
1530
-
1535
1540
1545
1550
Wavelength [nm]
-
1555
1560
_,50
900
..-.-..
....
-..
CL
8 50
800
1530
first
anomalo~~~us)emn
.
1535
ossigo
....
1540
.. .
- . . ..
. . .. .-.
. . .. . ..-.
1545
1550
1555
1560
[nm]
0k
Wavelengthl~v
Figure 4-3: Group delay measurements (top) and dispersion calculation (bottom) for
uTM
54
15000
1O
I
CO 10000
...... ....
0.
.................. .......
5000
0
U)
.... . ..
0
CU
ci:
-5000
1530
T
-C
1535
1540
1545
1550
Wavelength [nm]
1560
1555
480
.........
-..
.................
-..
...
.... -..
460
0.
CL 440
0
420
0
40
0I
1530
1535
1540
1545
1550
1560
1555
Wavelength [nm]
Figure 4-4: Group delay measurements (top) and dispersion calculations (bottom)
for second anomalous segment consisting of 25 km AllWaveTM.
-8000
-
6000
a)
---
- -
-a
-
4000
a)
-)
U)
(
I
1530
I -1535
-.
. . .. .
- -..
.....-.
. ...
....
...- . -. ---.
. . -......-. -..
-.
2000
I
- - ....
- ..
- .
--...........
-.
, --1540
I
I
I
1545
1550
1555
1560
I
1555
1560
Wavelength [nm]
E
180
C
-. ....
-.
170 .......-.
0
-.... -.
..
...
..-
-..-.-.-.-.-.M160
0
150
15 30
1535
1540
1545
1550
Wavelength [nm]
Figure 4-5: Group delay measurements (top) and dispersion calculations (bottom)
for third anomalous segment consisting of 10 km SMF.
55
20000
0n
CO
-
--CL
=
-
- -
-
-20000
0)
-
-
-
>-40000
I
-60000'
15 30
i
1535
I
1540
-- ,
1545
-
1550
- j
1555
I
---
1560
Wavelength [nm]
-1400
CL
C,
..........
-.
-1450
.....
.
-.. ....
....
....-..
C
-1500
-.....
..
..
-.
...... -.
..
-.
...
0.
-1550'
15 30
1535
1550
1545
Wavelength [nm]
1540
1555
1560
Figure 4-6: Group delay measurements (top) and dispersion calculations (bottom)
for normal segment consisting of 15 km DCF.
.- 4800
CO,
4600
'CD
-
-.
..... -......
.....
...
0)
cl 4400
cc
42I
1530
1540
1535
1550
1545
1555
1560
Wavelength [nm]
50
C
25 - . . . . . . .
. . . . . .. . . . . . .
-..
... .. ..
C
0
C,)
CO,
*0
0
-
-25
1530
-
-..
.
. . .... . . .
1535
.
1540
-.
. .......
1550
1545
Wavelength [nm]
1555
-
1560
Figure 4-7: Group delay (top) and GVD (bottom) calculations for entire fiber loop
consisting of 75 km AllWaveTM , 10 km SMF, and 15 km DCF. Net zero dispersion
is achieved at 1550 nm as designed.
56
The dispersion profile shown in Fig. (4-7) remains fixed throughout all the experiments. The path-average zero dispersion wavelength for this fiber loop is approximately 1550 nm (it is slightly higher). The dispersion map can be determined by
tuning the laser source to a particular wavelength. For these experiments, the transmitter is tuned at 1549 nm, which places the map near zero dispersion but in the net
normal dispersion regime. The total dispersion accumulated in one roundtrip (about
100 km) is about -0.2 ps/nm (recall this is the D parameter).
4.2.3
Other Measurements
To measure the loss in each fiber section, pulses from a tunable semiconductor laser
with a known average power are sent into each spool and the average output power is
detected by a power monitor. A list of the measured loss from each section is shown
in Table (4.1). Note that these measurements only pertain to fiber loss and do not
include connection loss between fiber spools.
Fiber Section
Anomalous 1
Anomalous 2
Anomalous 3
Normal
Loss (dB)
9.3
5.0
2.5
9.5
Table 4.1: Measured loss for each fiber section in loop.
The loss of the acoustic-optic (AO) modulators, used for switching optical pulses
in the recirculating loop, can be estimated by continuously sending an acoustic wave
so that output of the AO modulator is always the diffracted signal (which is defined
to be the ON state in these experiments).
The estimated insertion loss resulting
from the acoustic Bragg grating and fiber coupling is measured to be approximately
3 dBm.
To obtain better estimates of the fiber lengths, an optical time domain reflectometer (OTDR) is used. This instrument injects a pulse into a spool of fiber whose length
is to be measured and receives a fraction of the pulse power reflected at the opposite
57
end of the fiber (which is left unconnected). This technique measures the fiber length
to within the accuracy of the index of refraction specified for the fiber under test in
the OTDR. Note this technique can also measure the fiber loss since the signal power
is continuously monitored and splices or lumped losses are apparent. The lengths for
each fiber section is listed in Table (4.2).
Length (km)
50.1
25.0
10.2
15.1
Fiber Section
Anomalous 1
Anomalous 2
Anomalous 3
Normal
Table 4.2: OTDR measured length for each fiber section in loop.
Since the most essential element in this project is to control recirculating pulsetrain
power levels, it is useful to characterize the gain and saturation power of the EDFA's
used in the fiber loop. The only user interaction with the amplifier is turning a dial
to set the pump level. A method to determine the gain characteristics in the EDFA
is to vary the input power and measure the amplifier power output. A diagram of
this measurement technique is shown in Fig. (4-8). A semiconductor laser source wih
Laser
LIser ---
Power
Monitor
with
Attenuator
---- >EDFA
Bnps
BFidpes
Fle
oe
Monitr
oio
Figure 4-8: Experimental setup for EDFA transfer function measurements with amplifier pump level varied between 1 and 10.
adjustable average power is used. An attenuator is utilized to attain fine control of
the optical input power. The EDFA transfer characteristic is measured with amplifier
pump level varied between 1 and 10 as shown in Fig. (4-9). The EDFA begins to
saturate with an average input power of about 0 dBm at high pump levels. Note that
the saturation power does not increase linearly with increased amplifier pump level,
but higher amplifier pump level still leads to a higher amplifier saturation power level.
58
Measured EDFA gain characteristics for each pump level
15
I
I
I
I
I
I
10
5
0
m -5
0
CO
0
.-
15
-20
-25
x
-30
+
*
o
-35
-40
-35
-30
I
-20
-15
-25
Input average power [dBm
Pump Level 1.0
Pump Level 2.0
Pump Level 5.0
o
Pump Level 8.0
Pump Level 10.0
-10
-5
0
Figure 4-9: Measured amplifier gain and saturation power at different pump levels.
59
4.2.4
Recirculating Fiber Loop
A fiber loop consisting of the dispersion map described previously is used to simulate
long distance propagation by propagating pulses around the loop and measuring the
loop output gated by a modulator after a determined number of periods [27, 28].
While a transmission loop does not exactly describe an embedded terrestrial fiber
link, it is more feasible and economical for laboratory demonstrations.
The physical layout of our recirculating fiber loop is shown in Fig. (4-10). The
fiber loop corresponds to the designed dispersion map. Two EDFA's are used to
OPTICAL
OPTICAL
SWITCH
-
SWITCH
LASER
SOURCE
50/50 COUPLER
I
RF
BIAS
OPTICAL
SWITCH
DRIVERS
GATED
TRAIN
-
__
OFE
50 km
SWITCH
ALL-WAVE
A/B
A/B
DELAY/PULSE
GENERATOR
C/D
(RSF
35 km
ALL-WAVE
OR SMF)
15 km
DCF
EDFA
EDFA
Figure 4-10: Experimental setup of recirculating fiber loop
compensate for fiber loss with spacing arranged such that both EDFA's provide approximately equal gain. Two acoustic-optic (AO) modulators are used as switches to
control the loading and unloading of the data on the loop. When an acoustic wave is
injected in the path of the input beam, a Bragg diffraction grating is created and most
of the light is scattered into the first-order output. There is a relative 100 Mhz phase
shift induced by this grating but it is negligible compared to the bandwidth of the
optical pulses in the loop. In order to maximize extinction whenever the switch is off,
the first-order output (the diffracted signal) is used as the AO modulator output and
60
the zero-order output is left unused. One of the modulators is placed after the short
pulse laser source for loading purposes and the other is placed at the end of one loop
period for continuing or clearing the loop. The timing for both switches is controlled
by a single delay/pulse generator. The control signals into each modulator have an
inverse relationship. The loading switch closes to load pulses from the laser into the
loop while the other switch opens to clear remaining pulses from the previous cycle.
When the loading switch opens, the pulses propagate continuously around the loop.
The rate at which the loading switch opens and closes affects how many periods one
set of optical pulses can cycle around the loop because a new "experiment" begins
with a new set of pulses. The loading time window is 500 ms, which is one period
around the approximately 100 km loop, so that pulses can fill the whole loop. If the
rate of the loading switch is 50 Hz where new pulses are loaded every 20 ms, then
pulses can propagate up to 40 periods (i.e about 4000 km) around the loop.
In order to view the pulses in progress around the loop, they must be gated by a
modulator. An AO modulator is used to gate the pulses after a determined number of
loop periods in Fig. (4-10). The RF signal driving the AO modulator is output from
the same delay/pulse generator used to drive the loading/clearing of the loop. The
time of the delay from the loading of the loop can be programmed in the generator
and this delay tells when to look at the pulses after a certain number of periods in
the loop. The gated pulses can then be analyzed by diagnostics equipment such as
sampling oscilloscope, autocorrelator, or optical spectrum analyzer. Signal loss due
to incomplete switching in the AO modulator is mitigated by amplifying the signal
before the AO modulator. As an experimental note, it is preferable to place the EDFA
before the modulator so that noise occurring outside the gated window will not be
amplified to deteriorate the signal-to-noise ratio (SNR) at the diagnostic equipment.
4.3
Experimental Results and Discussion
Experiments on the recirculating fiber loop reveal the possibility for the existence of
periodically stationary pulses at relatively low energies. A phenomenon observed in
61
both preliminary numerical simulations and experimental work is the shifting of the
transform-limited pulse state position because of the net normal dispersion imbalance
in the map. In the linear case, dispersion alone cannot mitigate this shifting. In the
presence of nonlinearity, however, experiments indicate that sufficient power levels
can help compensate for the walking of the minimum pulse width position. Fiber
loss and lumped amplifier gain complicate the dynamics of this shifting.
Third-
order dispersion presents an additional challenge since the direction of motion in the
minimum pulse width position shifting becomes dependent on launch position and
power levels. Initial location, chirp, and power launch conditions therefore need to be
"right" in order to avoid unnecessary transient states and to approach a periodically
stationary steady state.
4.3.1
Preliminary Observed Effects of Power Level
The recirculating fiber loop is initially set up to launch laser source pulses between
the 50 km and 35 km sections of anomalous fiber such that the launch point is located
7.5 km before the half-cell center of the anomalous segment. A diagram of this loop is
shown in Fig. (4-11). The optical amplifiers are placed approximately at the middle
of each dispersion half-cell segments so that the pulses are amplified when they are
near their transform-limited state. Placement of the EDFA's at the half-cell center
improves the possibility of observing stable pulses than placement at the half-cell
boundaries where the pulses tend to have the broadest pulse widths and intersymbol
interference (ISI) occurs. Amplifying ISI at the half-cell boundary may introduce
other nonlinearities and enhance cross-phase modulation. Studies have shown that
judicial placement of amplifiers in dispersion-managed fiber links may be critical to
stable DM soliton propagation [29, 30]. An inline filter with a passband of 2 nm
follows each EDFA in order to eliminate most of the ASE noise. Because the filter
bandwidth is much broader than the pulse's bandwidth, the temporal broadening due
to filtering some side spectral components is not significant, especially when breathing
62
of DM solitons causes the pulses to be much wider than the launched pulse width. 1
TAP
LAUNCH
50 km
ANOM
35 km
ANOM
6 km
NORM
9 km
NORM
Figure 4-11: Dispersion map with a tap at launch point located 7.5 km before the
anomalous half-cell center.
The gated output pulsetrain is directed to an autocorrelator where the pulse width
is accurately measured. The autocorrelator provides a useful diagnostic tool for analyzing these gated pulses since it does averaging over a selected integration time span
such that the gating effect becomes transparent. Although phase synchronization of
a digital sampling scope is not straightforward at the 25-50 Hz loop output gating
frequencies, this device still provides some use as a diagnostic tool. The main use of
the scope is to determine whether pulses can still be observed after many roundtrips
in the loop. This technique is more time-efficient than waiting for a slow scan in
the autocorrelator since pulses can be effectively viewed in "real-time" on the scope.
An optical spectrum analyzer (OSA) also provides a useful diagnostic tool but synchronization must be achieved between the loop output gating signal and the OSA
spectral sweep measurement speed. This synchronization is achieved by externally
triggering the OSA with the gating signal originating from the delay/pulse generator.
The primary use of an OSA in our recirculating loop characterization is to observe
nonlinear effects, which can be inferred by spectral changes seen in the OSA.
'Studies in [31, 32] have shown that very narrow filtering can make a difference in establishing
stable pulse propagation.
63
The first experimental objective is to observe the effects of changing power levels
in the loop by adjusting the initial launch average power or the pump levels of the
EDFA's. The goal is to produce periodically stationary pulses after a few roundtrips
given the proper power settings based on results from numerical simulations. The
pulses from the loop are first analyzed at the launch point. The evolution of the
pulses around the loop is tracked by viewing them after the first few roundtrips (usually ten to fifteen) and the stability is inferred from the results after "long-distance"
propagation (about twenty to thirty periods around the loop). In our recirculating
loop experiments, we maintain a fixed initial launch power because the modelocked
fiber laser output power going into the loop does not play as big a role as the power
levels within the loop governed by the EDFA's. Keeping the laser power constant
throughout all experimentation may introduce additional transients as the pulses try
to find their periodically stationary state in the dispersion map because of possible
power mismatch between the initial state and steady state but the amplifiers are generally saturated after the first couple of periods in the loop and transients disappear
after a while provided the amplifier pumping powers are set judiciously. The main
variables in the loop experiments are thus the gain levels of the EDFA's with the
average launch power set to 1 mW or 0 dBm (the actual laser output is 6 dBm, but
we suffer a 3 dB loss loading the AO modulator and an additional 3 dB loss from the
50/50 coupler at the launch point in the loop).
The pulse width evolution with the number of roundtrips through the fiber loop
is shown in Fig. (4-12). On this plot, the pulse widths are recorded after a particular
roundtrip (one loop period is approximately 100 km) for two distinct power levels
labeled "high" and "low." The "high" power level corresponds to the loop EDFA
outputs, which provide about an order of magnitude higher power compared to the
"low" power level (i.e. "high" 10 mW vs. "low" 1 mW). In this initial experiment,
the absolute magnitude of the power is not as crucial as the relative power levels
being compared. As an aside, the "low" power setting has an estimated average loop
power of less than 1 mW, from which we can assume occupies the low energy regime
for a dispersion-managed soliton with an unchirped pulse width of 5 ps. Fig. (464
Pulse width measurements at launch point
50
- a, LOW power
-4-
45-
HIGH power
640-
35 -
30 All
025'I)'
C
02
202(,I'
A
0
0
0
05
15
10
20
25
30
Number of periods around loop
Figure 4-12: Experimental data of pulse width evolution versus loop period measured
at the launch point in the fiber loop.
12) indicates that changing the power levels leads to apparently different broadening
of the pulses when measured at the launch point. Rather than interpreting this as
simply pulse broadening, however, we theorize that the position of the minimum
pulse width (or transform-limited pulse state) is walking away or slipping from the
launch point where we measure the pulses. Since we cannot track this minimum pulse
width position as it "travels" around the loop, this shifting of position is translated
to increased pulse broadening at a fixed point of observation as the transform-limited
state moves away. It is clear from the data in Fig. (4-12) that the power level can
potentially alter the dynamics of this shifting of the minimum pulse width position.
65
4.3.2
Shifting of the Minimum Pulse Width Position
The cause of this apparent shifting of the transform-limited pulse state position is
simple if we consider an ideal case in which we include dispersion (GVD) but ignore
loss and nonlinear effects. If the dispersion map has exactly zero net dispersion and
a transform-limited pulse is launched into it, the pulse always returns to its original
pulse width at the launch location. This holds true regardless of where the pulse is
launched. If, however, the linear map has some net dispersion, the chirp-free point
walks away from its initial point at a rate that is proportional to the dispersion
imbalance, as illustrated in Fig. (4-13). If the dispersion map lies on the net normal
Later
0.
A
1st
LAUNCH
35 km
50 kmn
ANOM
ANOM
9 km
NORM
6km
NORM
Figure 4-13: Illustration of how the tranform-limited state position shifts in the linear
case with negligible loss and nonlinear effects.
dispersion regime, every roundtrip in the loop picks up a net normal dispersion and
the pulse needs to travel a little farther through anomalous fiber (if the launch point
66
occurs in the anomalous dispersive segment) in order to compress back to its chirp-free
state. Note that the motion is backwards, i.e. against the direction of propagation, if
the pulse is in the normal dispersive segment since normal fiber must be "subtracted"
to retain the minimum pulse width with a net normal dispersion. In the linear case,
this shifting of the minimum pulse-width position is an unrecoverable process since
nothing can compensate for a non-zero net dispersion.
Nonlinearity in optical fiber can help counter the dispersion-induced shifting of
the minimum pulse width position. Exactly analogous with a regular soliton, the
right amount of nonlinearity will balance the net dispersion associated with the map
and thus completely halt the shifting. While the dispersion-managed pulse continues
to breathe temporally due to the dispersion swings in the map, proper power levels
can establish a periodically stationary pulse condition where the transform-limited
state returns to the point where a chirp-free pulse is launched (assumed to be the
center of either dispersion segments). This analysis, however, assumes that the energy
is conserved. Real fiber is lossy in addition to being dispersive and nonlinear. The
dynamics of the shifting of the minimum position is much more complicated in the
presence of attenuation since the nonlinearity changes according to the pulse intensity.
Most of the propagation in the loop can be approximated as linear since most of the
pulse energy is attenuated by loss in the fiber. Almost all of the nonlinear action
occurs right after the EDFA's since this is where the peak power is highest. Note
that in this fiber loop, most of the nonlinearity occurs in the anomalous segment
because peak intensity in normal segment is lowered due to quicker spreading from
a much higher local dispersion. We can argue that the transform-limited pulse state
position lies within the nonlinear regions after amplifiers and not necessarily exactly
in the center of the segments as described by the lossless case. A plausible reasoning
for this is the following: if we invoke the principle of minimum average energy, the
minimum pulse width should occur where the peak intensity is the highest, which
occurs in the most nonlinear region. This is analogous to the variational approach
in seeking the lowest energy state (i.e. ground state) in a lossless system. Of course,
the chirp-free point (not equivalent to transform-limited) may not be defined in some
67
dispersion maps and so the launched pulses may need to be chirped properly in order
to avoid persistent transient states. In any case, neglecting noise, the behavior of the
minimum pulse width position shifting is primarily controlled by the amplifier pump
levels after a certain number periods around the loop.
The data shown in Fig. (4-12) indicate that the higher-power level leads to a
quicker rate of broadening, which implies a faster rate of shifting of the minimum
pulse width position.
This is rather puzzling since numerical simulations predict
that from the linear case, adding some nonlinearity tends to slow down this rate of
shifting as it counters the net dispersion in the map. Possible explanations exist,
however, for this counterintuitive phenomenon.
It may be that the "high" power
level is too high such that excess nonlinearity contributes to more pulse spreading in
addition to the dispersion. However, the trend of increased gain from the amplifiers
leading to increased broadening is observed from the "low" power to the "high"
power level and it is hard to imagine that the "low" power setting has too much
nonlinearity, especially since a lower power level than this cannot overcome the loss
in the system. An alternative possibility is that initial conditions may be set in such
a way that more nonlinearity creates further broadening. As mentioned in Chapter
3, launch conditions are a determinant for stable dispersion-managed soliton systems
and mismatched launching parameters at the launch point in the dispersion map may
never lead to a stable steady state. Since the initial chirp, is not controlled due to
experimental difficulty in choosing the "correct" fiber length to prechirp the laser
pulses via a trial-and-error fashion and to the lack of a "tunable" chirping device, a
periodically stationary pulse may never emerge from the loop setup depicted in Fig.
(4-11). Nonlinearity therefore cannot alone resolve the chirp mismatch and may even
accelerate pulse broadening to the point where the minimum pulse width location
cannot be realized in the map any more. The "new" transform-limited pulse state,
however, continues to shift position at a rate governed by the power levels in the loop.
A more in-depth numerical model is required to justify effects of launch conditions
on the dynamics of the pulses in the loop.
68
4.3.3
Robustness of Pulses After Long-Distance Propagation
Despite the presence of this apparent shifting of the minimum position, the pulses still
remain intact after "long-distance" propagation. Fig. (4-14) shows the autocorrelation traces for pulses with the same two "high" and "low" power levels represented
in Fig. (4-12) after about 3000 km (30 periods around the loop). Note that these
6000
5000
5000-
4000-
4000
FWHM=
FWHM=
- 3000
*i-,300030-
21 psA*
-22
ps
4-
4-~
2000 -
-2000
1000-
1000
0
0 10
-50 -40 -30 -20 -10
Time [ps]
20
30
40
50
0
10
0
-50 -40 -30 -20 -10
Time [ps]
20
30
40
50
Figure 4-14: Autocorrelation of pulses after 3000 km propagation for "low" (left) and
"high" (right) power levels with 6.7 km SMF anomalous loop compensation fiber.
results are obtained after compensating the broadened pulses after 30 roundtrips with
some anomalous fiber. While one may view this as compensating some of the total
normal dispersion picked after several periods of the dispersion map with anomalous
fiber, one may also interpret the placement of anomalous fiber after the loop as effectively "moving" the tap point beyond the launch point to counter the minimum
position shifting. In this experiment, we use 6.7 km of compensating anomalous fiber
to compress the broadened pulses as shown in Fig. (4-14).
The results shown in Fig. (4-14) are not necessarily the transform-limited pulse
widths since the data pulse widths of 15 ps is substantially longer than the 5 ps pulse
width of the launched pulses. It is possible, however, that the transform-limited pulse
width for the dispersion map used in this experiment is longer than 5 ps. Seeking this
transform-limited pulse width is nontrivial, since various taps must be implemented
around the loop to detect pulse widths at those points and infer a local minimum from
69
this set of data fitted to some hyperbolic curve. Even if tapping the loop at several
points without adding loss were possible, measuring the pulse widths might still be
impossible because the ultrashort launched pulses spread very quickly due to a high
map strength with a strong dispersion swing and very short pulse width. At certain
taps, especially those around the boundaries of the segments, pulse broadening may
be so great that multiple pulse ISI renders autocorrelation of pulse widths useless.
The goal of the autocorrelation traces presented in Fig. (4-14) is to illustrate the
survivability of the dispersion-managed pulses after many periods around the loop
even at relatively low power levels.
4.3.4
Achieving Periodically Stationary Pulses
In order to analyze the shifting of the minimum position more closely, we alter the
fiber loop setup in order to initially launch pulses 7.5 km after the center of the halfcell and tap the loop about 1.7 km after this initial launch point. The restructured
loop setup (with the same dispersion map) is depicted in Fig. (4-15). Since this 35
km span actually includes the fourth anomalous section consisting of various spools
of SMF-28 discussed above, the loop can easily be tapped at the connection points
between the spools. The tap merely consists of an 80/20 coupler or splitter where
the 80% of the power returns into the loop while 20% goes into the gating AO switch
(the coupling loss is negligible compared to the accumulated fiber loss).
As predicted from numerical simulations, we assume the motion of shifting is in the
direction of propagation. In this case, at the tap point located 1.7 km from the launch
point, the measured pulse width will initially compress as the minimum position moves
towards it and then broaden when the minimum position passes through the tap
point and continues to move away from it. The measurement during the first period
should remain constant for any power levels set by the pumping of the amplifiers
since the initial average launch power is kept fixed for experimental simplicity. The
first pulse width measurement should ideally be that of a pulse having dispersed
in the distance between the launch and tap point. Of course, if the pulses achieve
some periodic stationary state, then their pulse widths (and shapes and energies)
70
TAP
LAUNCH
35 km
ANOM
50 km
6 km
ANOM
NORM
9 km
NORM
Figure 4-15: Dispersion map with launch point located 7.5 km after the anomalous
half-cell center and a tap located 1.7 km from the launch point.
remain approximately the same at the fixed point of observation during all later
periods (neglecting the buildup of amplifier noise after every roundtrip). It must be
noted that no guarantee exists that the transform-limited pulse width position will
be measured at the tap point for any period. We can only infer that this position is
shifting through the tap point where a local minimum can be defined at some period
in the measured data or that this position enters some stationary state where the
pulse width measurements at the tap point approach a constant value.
The data acquired from this experimental setup is shown in Fig. (4-16). The autocorrelation pulse width is plotted for each of the first several loop periods. Again,
the initial launch power is fixed (at 1 mW) such that the pump levels of the EDFA's
control the average loop power. The pump levels of EDFA's are set at 8.0 for the
"high" power level and at 2.0 for the "low" power level such that "high" is about
an order of magnitude larger than "low." If we consider the lower power curve (and
approximate it as the linear case given that lower gain from the amplifiers cannot
compensate for the fiber loss sufficiently), we see the pattern of the minimum pulse
width shifting through the tap point. Initially, the pulse width shortens as the minimum position approaches the tap point. The data curve reaches a local minimum
as the transform-limited position passes the point of measurement. Then the pulse
71
Pulse width measurements off launch point
I
SI
I
I
I
I|
a - LOW power
30
-e- HIGH power
28
~24
U_
-522
'I I
7520
0.
C
0016
0
',
12
I
10
2
4
I
I
6
8
13II
12
10
Nth period in loop
I
14
16
18
20
Figure 4-16: Experimental data of pulse width evolution versus loop period measured
at the loop tap point
width starts to broaden as the minimum pulse width position walks away. The aberrant initial steep gradient for the pulse width compression observed in the data is
most likely due to the transients since the initial conditions (including launch power
and chirp) remain unadjusted.
While the lower power curve has the persistent shifting of the minimum position
as expected in the linear case, the higher power curve demonstrates that nonlinearity
can actually decrease the rate of this shifting. As seen in Fig. (4-16), the minimum
position appears to move more slowly towards the tap point on the "high" power
curve (inferred from a slower rate of pulse width broadening measured at the tap
point). After a few periods around the loop, the pulses with the "high" power level
then approach a quasi-steady state where the pulse widths remain approximately
constant for several loop periods. This suggests that the pulses may have reached
72
a periodically stationary condition before they start to broaden again much further
away perhaps because of signal degradation by amplifier noise accumulation. Note
that this "stable" pulse width is not necessarily the transform-limited state since the
minimum pulse width position may occur at a point different from the tap point
during the establishment of a periodically stationary state in the fiber loop. Similar
to the pulses with "low" power, rapid changes occur in the first few loop periods
because of transients due to mismatched launch parameters. However, it is apparent
that the rate of minimum position shifting is much lower in the "nonlinear" pulses
(with "high" power) than in the "linear" pulses (with "low" power) in Fig. (4-16).
Hence, this is experimental indication that increased optical power levels help to
compensate for the dispersion-induced shifting of the minimum pulse width position
during DM soliton propagation in a recirculating fiber loop.
4.3.5
Discussion of Experimental Results
A comparison between Figs. (4-12) and (4-16) shows that launching conditions are
crucial in establishing a periodically stationary state. In Fig. (4-12), it appears
that increased powers inducing higher nonlinearity actually exacerbate the dispersioninduced shifting of minimum pulse width position rather than compensating for it.
The suspected cause of this counterintuitive behavior may be mismatched initial
conditions, which may involve chirp and third-order dispersion. If this were the case,
we should still observe a steady-state trend when larger nonlinearity starts to slow
down the rate of shifting when the transients from random launch parameters die
away. The builtup of ASE noise, however, can prevent observation of the steadystate because of a severely degraded SNR. The increasing pulse broadening at the
observation point may be due to the degraded signal's inadequate intensity to excite
nonlinearity needed to compensate for the dispersion. In Fig. (4-16), the occurrence
of transients is still apparent in the sharp changes of pulse width on both curves.
However, the initial conditions may be set in such a way that we are able to see how
nonlinearity can mitigate the effects of the minimum position shifting before noise
becomes dominant many loop periods later. Further experimentation, therefore, needs
73
to be done where initial conditions need to be somehow set "correctly" to avoid as
much transient evolution as possible to observe a periodically stationary state before
amplifier noise takes over. If setting the initial parameters to the "right" values is
infeasible due to equipment limitations, then the noise problem would need to be
addressed so that the pulse can survive much longer in the loop to be measured and
analyzed.
In any case, the two sets of data reveal two properties in actual fiber systems.
Fig. (4-12) illustrates that the presence of gain and loss modifies the dynamics of
the minimum position shifting compared to the constant rate in a linear system with
a dispersion imbalance. Fig. (4-16) provides evidence that a periodically stationary
state can be achieved if the initial conditions are favorable and if sufficient power levels
are available for the nonlinearity to push back the shifting of the minimum pulse width
position. The seemingly contradictory behavior between the two sets of data may
imply some metastability associated with balancing the dispersion-induced shifting
and fiber nonlinearity. If the launch conditions are not "right," then increasing the
powers can lead to divergent motions, where one pushes against the minimum position
shifting while the other appears to enhance it. We suspect that third-order dispersion
may be responsible for this reversal of direction because of the asymmetry associated
with the point of observation.
Because of fast broadening over just a short distance of propagation due to a
very strong map, the nonlinear action takes place over a small region right after the
amplifier since the peak intensity reduced by intense dispersion does not excite enough
nonlinearity. In fact, most of the nonlinearity occurs in the anomalous dispersion
segment because of the high dispersion in the normal segment. If the net dispersion in
the map is normal, the shifting of the transform-limited state moves in the direction of
propagation as explained previously. Since most of the nonlinearity obviously occurs
after the EDFA's (and over a longer distance in the anomalous segment near the
launch point), this shifting has a chance of being pushed back in the presence of
adequate nonlinearity provided that the launch conditions are good. An interesting
observation that can be made from these results is that transmission of the lower74
energy dispersion-managed solitons can be regarded as mostly linear propagation
perturbed by the nonlinear Kerr effect. Much of the propagation can be approximated
as linear because of the low pulse intensities due to fiber loss and rapid spreading
(characteristic of a high map strength). When the pulses hit an amplifier, however,
the pulse intensity shoots up and there the nonlinearity takes effect. If the amplifier
gain is adequate, the nonlinearity can provide the right kicks to knock the broadened
pulses back to their transform-limited shape such that the minimum position shifting
can be stopped to produce a periodically stationary state.
75
Chapter 5
Conclusion and Future Work
Preliminary experimental evidence indicates that dispersion-managed solitons on the
lower-energy branch in the path-average normal dispersion regime can exist provided
the initial launch conditions are proper and the power levels are sufficient. Numerical
studies show that a continual, unrecoverable shifting of the transform-limited pulse
state position occurs in the absence of nonlinearity if only group velocity dispersion is
considered and a dispersion imbalance exists in the fiber loop. Nonlinearity, however,
can compensate for this dispersion-induced shifting in a way that is analogous to a
regular soliton's balance between dispersion and nonlinearity. The loss and "lumped"
gains associated with a real fiber system make the dynamics of this minimum position
shifting more complicated. Essentially, most of the nonlinear action occurs after the
amplifiers where the pulse intensity is highest. These "nonlinear regions" therefore
provide the "kicks" to push back the shifting by fighting the dispersive pulse broadening. Experiments reveal that the launch parameters are crucial since increased power
(to provide more nonlinearity) can have either a beneficial or detrimental effect. If
the launch conditions are favorable, however, sufficient nonlinearity can effectively
decrease the rate of minimum position shifting to produce periodically stationary
pulses. This effect is shown in the experiment where pulses measured at a tap point
located away from the launch point approach a quasi-steady state after transients
disappear a few periods around the recirculating fiber loop. The average operating
powers in the loop are relatively low compared to numbers cited from typical non76
linear dispersion-managed systems. For this reason, we believe we have observed
low-energy DM soliton propagation in a recirculating fiber loop. The key idea is that
most of the propagation observed in these experiments is mostly linear but perturbed
by the nonlinear Kerr effect.
The possibility of establishing periodically stationary solitary waves with low energies opens up many avenues for future research. One future plan is to investigate
the shifting of the transform-limited state position in more detail. Whether or not
there exists a "periodicity" of position in concert with the periodicity of the dispersion
map remains to be seen. There appears to be some evidence from the experiments in
this thesis that pulses remain robust even in the presence of the minimum position
shifting and this can be explored further. Of course, rigorous modeling of the system
(including noise and higher-order dispersion) may be needed to establish the proper
initial conditions for launching pulses into a particular dispersion map and to maintain stable steady-state propagation where the nonlinearity works to achieve a stable
minimum pulse width location.
Research on dispersion-managed solitons has convincingly shown that they can
resolve some problems associated with current practice. Not limited by homogeneous
dispersive fiber, DM solitons can exist in all dispersion regimes, which is ideally
suited for wave-division multiplexing (WDM) systems. As mentioned in this thesis,
existence in the path-average normal dispersion regime reveals two branches of energy
solutions. The pulses on the lower-energy branch require less power, a desirable factor
in practical systems. In addition, this quasi-linear branch is close to net zero dispersion, where timing jitter such as the Gordon-Haus effect is substantially reduced.
Operation around net zero dispersion with a local dispersion high enough to support
the DM soliton's energy mitigates jitter that plagues regular soliton systems [33, 34].
Nevertheless, DM solitons should not yet be regarded as the "silver bullet" to optical
fiber communications. It may be argued that they do not deserve the appellation
"soliton," which implies a "particle-like" nature, because they cannot readily survive
pulse-to-pulse interactions or collisions [4, 35, 36].
Higher-energy pulses consume
the lower-energy ones in a one-to-one collision [4]. While DM solitons can still be
77
considered superior to linear techniques (i.e. NRZ) in combatting polarization-mode
dispersion, it may still be debatable on how they perform against PMD compared to
regular solitons [37, 38].
In any case, the lower-energy DM solitons investigated in this project may potentially earn a competitive edge in the commercial market due to their favorable
low energy/power requirements and quasi-linear nature. When these optical pulses
can be reliably produced with better control of the initial launch conditions and
power levels in fiber links, characterization experiments such as timing jitter, collisions/interactions, and higher-order dispersion can be done to determine whether
lower-energy DM solitons operating in the quasi-linear net normal dispersion regime
are suitable and desirable in optical fiber communication systems.
78
Appendix A
Numerical Simulation of
Experiments
The symmetric split-step Fourier method is used to numerically simulate pulse propagation in the recirculating fiber loop with the dispersion map specified above. The
algorithm is applied in a stepwise fashion for each of the dispersion sections to account
for dispersion management. The FFT window length is chosen to be wide enough
(about 1500 ps) to accommodate the massive pulse spreading due to a very strong
map.
Our numerical modeling assumes a Gaussian-shaped pulse envelope that is to be
launched into the dispersion map. Initial conditions, such as average power and pulse
width, can be specified. While the dispersion profile of the fiber loop is fixed with
the fiber types that are used, the map and the net dispersion are set by specifying
the center wavelength.
Fiber loss and the nonlinear coefficient in the NLSE are
specific to the fibers and thus remain constant for all simulation trials. The gain
for the EDFA's, treated as lumped elements, can vary by adjusting the pump power
level in accordance with the gain characteristics measurements shown in Fig. (4-8).
Amplifier power saturation based on measurements is modeled in the simulations.
For simplicity, optical filtering is not simulated. Since the fiber-coupled filters used
in the loop have a broad bandwidth relative to the pulses, the filtered pulses are not
significantly affected. Noise analysis is also omitted in our simulations for simplicity.
79
While modeling ASE noise is nontrivial because of its stochastic nature, it is possible
to simplify noise as some constant background added to the signal. However, since
we are only interested in finding the proper initial conditions for launching a pulse
into the loop, we ignore noise so it does not interfere with the search for steady-state
pulses.
Numerical simulations on the recirculating fiber loop used in this project are initially done to determine whether periodically stationary pulses exist. Preliminary
numerical simulations assume lossless fiber and omit lumped amplifier gains. In addition, transform-limited optical pulses are launched in the middle of the segments and
no prechirping is required. Under these assumptions, the only adjustable parameter
in this ideal fiber loop simulation is the average launch power for a given operational
center wavelength. Once the optical pulse power needed to sustain a steady state is
found, loss and lumped amplifier gains are added. In this more realistic loop simulation, amplifier pump level is introduced as a means of controlling the average loop
power. The pump level of the amplifier is specified and delivers the gain according
to the EDFA measurements in Fig. (4-9). Attenuation can also be specified to account for fiber insertion loss or lumped connector loss. Initially, the transform-limited
pulses are launched in the middle of the half-cell to determine the transmitter launch
power and amplifier pump level required to support periodically stationary pulses.
Next, transmitter pulses are injected into the loop half-cell off-center to determine
the effects of initial chirp mismatch and to see whether pre-chirping using fiber is
required to produce a steady state. At this point, we have a full numerical simulation
of the actual loop used in our laboratory experiments. Later, third-order dispersion
is included in the simulation to analyze the effects of asymmetry, which may play a
crucial role on the launch location when short optical pulses are injected into a strong
dispersion map.
Some simulation results plotting FWHM pulse width versus loop period are shown
in Figs. (A-1, A-2, A-3). The fiber loop setup is the same as that depicted in Fig.
(4-15). In these numerical simulations, initial conditions (chirp and average launch
power) are kept fixed for simplicity, which means transients will most likely occur. In
80
each plot, three points of observation (the launch point and two points tapped before
and after the launch point) are shown to analyze the transform-limited state position
shifting more closely. Fig. (A-1) illustrates the linear case. Fig. (A-2) plots the full
simulation (dispersion, nonlinearity, loss, and gain) with just enough amplifier gain to
compensate for fiber loss. Fig. (A-3) shows a simulation that adds more nonlinearity
with a higher average power level. These plots show that nonlinearity can slow down
the dispersion-induced shifting of the minimum pulse width position and lead to a
periodically stationary state.
45
- -
Launch
Later
.Tap
Tap Earlier
40
35
-
-
-
-
30
C2
c0-25
15
10
0
5
10
15
20
Number of loop periods
25
30
Figure A-1: Simulation of the linear case with only dispersion and no nonlinearity or
loss.
81
9A
22-
Launch
Tap Later
.
- - Tap Earlier
20
18
16
~14
-3
12
U1
10
8
6
4
0
5
10
15
20
25
30
Number of loop periods
Figure A-2: Simulation of fiber loop with enough amplifier gain to compensate fiber
loss.
-
Launch
-.-.- Tap Later
12 - - - Tap Earlier
11
W
-
-
10-
-C
a,
9
M)
876
0
5
10
20
15
Number of loop periods
25
30
Figure A-3: Simulation of fiber loop with a higher pump level to show that nonlinearity can mitigate the shifting of the minimum pulse width position.
82
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