Dispersion-Managed Solitons in the Path-Average

Normal Dispersion Regimeby

Samuel Tin Bo WongSubmitted 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 anddistribute publicly paper and electronic copies of this thesis document BARKER

in whole or in part. MASSACHUSETTS INSTITUTEin woleor n prt.OF TECHNOLOGY

JUL I120

LIBRARIESAuthor ... . .. . .. .... . . . . . . . ....................

Department of Electrica Engineering and Computer ScienceMay 23, 2001

Certified by .............................. ..............Hermann A. HausInstitute ProfessorThesis Supervisor

C ertified by ..... .. ................ .............................Scott A. Hamilton

MIT Lincoln Laboratory Staff4hes ,*Supervisor

A ccepted by ............. ............... .. . ....................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 Scienceon May 23, 2001, in partial fulfillment of the

requirements for the degree ofMaster 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 transmittingdata via fiber because of the balancing effects of linear anomalous group velocitydispersion (GVD) and nonlinear intensity-dependent self-phase modulation (SPM).Regular soliton systems (with uniform anomalous fiber), however, have yet to entercommercial markets because of high power requirements and timing jitter. Anotherclass of solitary waves called dispersion-managed (DM) solitons, which can be man-ifested via a dispersion map, can resolve these issues. Numerical simulations andvariational methods have shown that DM solitons can propagate in the path-averageanomalous, normal, and zero dispersion regimes because of the interplay involvingdispersion, nonlinearity, and chirp. In the net normal dispersion regime, a plot ofpulse energy versus net dispersion yields two theoretical energy branches with mapstrength 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 zeromeans reduced timing jitter. The lower-energy DM solitons also need less power butexploit enough nonlinearity to fight GVD. So far, no one has claimed to have actu-ally found these pulses. This thesis provides preliminary experimental evidence forthe existence of these lower-energy DM solitons. Numerical and experimental studiesshow a shifting of the transform-limited state position due to a dispersion imbalance.Under the proper initial launch conditions, nonlinearity can mitigate the effects ofthis 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. HausTitle: Institute Professor

Thesis Supervisor: Scott A. HamiltonTitle: MIT Lincoln Laboratory Staff

2

Acknowledgments

I would first like to thank Prof. Hermann Haus for his guidance and mentorshipthroughout my thesis project. I am forever grateful not only because he introducedme to an exciting area of research but also because he gave me some much-neededdirection when my Lincoln group was suffering a mass exodus with people leaving foroptical startup companies at that time. Like every one of his students before me, Iam very honored to have Prof. Haus as my advisor. I would like to very much thankScott Hamilton for acting as my Lincoln supervisor despite his having many othertime-consuming duties in the Lab. I greatly appreciate his experimental advice andcareful 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 previousLincoln supervisor, for showing me how to construct a recirculating fiber loop. I thankJohn Moores for answering my (naive) questions on dispersion-managed solitons whenI first started to study them. I thank Prof. Erich Ippen for taking some of his valuabletime to discuss my experimental setup and results and providing some very helpfulinsight (I wished I had more opportunities to talk with him). I thank Tom Murphyfor his gracious help in the dispersion measurements and numerical simulations. Ithank Leaf Jiang, my officemate on campus, for expressing interest in my thesis andfor 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 Savagefor letting me time-share the modelocked fiber laser (and providing some data forthe laser). And I would like to thank Todd Ulmer for being cool and for introducingVan Halen to me so I can add punk, er, rock music to my "high-brow" repertoire ofBach, Beethoven, and Mozart (I need my daily dosage of "Hot for Teacher" to prepme up for labwork). Finally, I wish to extend my thanks to anyone in my Lincolnand 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 anyacknowledgments section of a typical dissertation. Well, I'm not going to do that notbecause I don't care about them but because it seems so cliche. I know my familysupports me no matter what endeavour I undertake and for me to openly thank themwould be superfluous. Sincerity is all that matters. I would, however, like to take thisopportunity to thank my support network of friends at MIT, especially those whomI met during my senior year. I guess I should also acknowledge someone at MIT (sheknows who she is) for allowing me to realize that there's so much more to life thanjust math and physics. But I'm not going to talk about my philosophy on life rightnow. I shall reserve that for the acknowledgments section of my doctoral thesis.

3

Contents

1 Introduction

2 Background on Fiber Properties and Regular Solitons

2.1 Intrinsic Fiber Properties . . . . . . . . . . . . . . . . . . . . .

2.1.1 D ispersion . . . . . . . . . . . . . . . . . . . . . . . . .

2.1.2 Frequency Chirp . . . . . . . . . . . . . . . . . . . . .

2.1.3 Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . .

2.1.4 Attenuation . . . . . . . . . . . . . . . . . . . . . . . .

2.2 Wave Propagation in a Nonlinear Medium . . . . . . . . . . .

2.3 Discussion on Regular Solitons . . . . . . . . . . . . . . . . . .

2.3.1 The Nonlinear Schrbdinger Equation . . . . . . . . . .

2.3.2 Properties of Regular Solitons . . . . . . . . . . . . . .

2.3.3 Limiting Factors for Regular Soliton Optical Networks

10

13

13

13

15

16

. . . . 19

. . . . 19

. . . . 22

. . . . 22

. . . . 26

. . . . 28

3 Background on Dispersion-Managed Solitons

3.1 Introduction to Dispersion-Managed Solitons . . . . . . . . . .

3.2 Methods for Theoretical Analysis . . . . . . . . . . . . . . . .

3.2.1 Numerical Simulation: The Split-Step Fourier Method

3.2.2 Approximate Method: The Variational Approach . . .

3.3 Behavior and Characteristics of DM Solitons . . . . . . . . . .

31

31

35

35

39

44

4 Experimental Search for Lower-Energy DM Solitons

4.1 Experimental Objective . . . . . . . . . . . . . . . . . . . . . . . . .

48

49

4

4.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . .

4.2.1 Laser Source . . . . . . . . . . . . . . . . . . . . . . . .

4.2.2 Dispersion Map and Measurements . . . . . . . . . . .

4.2.3 Other Measurements . . . . . . . . . . . . . . . . . . .

4.2.4 Recirculating Fiber Loop . . . . . . . . . . . . . . . . .

4.3 Experimental Results and Discussion . . . . . . . . . . . . . .

4.3.1 Preliminary Observed Effects of Power Level . . . . . .

4.3.2 Shifting of the Minimum Pulse Width Position . . . . .

4.3.3 Robustness of Pulses After Long-Distance Propagation

4.3.4 Achieving Periodically Stationary Pulses . . . . . . . .

4.3.5 Discussion of Experimental Results . . . . . . . . . . .

5 Conclusion and Future Work

A Numerical Simulation of Experiments

5

. . . . 49

. . . . 50

. . . . 51

. . . . 57

. . . . 60

. . . . 61

. . . . 62

. . . . 66

. . . . 69

. . . . 70

. . . . 73

76

79

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. . . . . . . . 27

2-2 Comparison between NRZ and soliton transmission formats for the

given data stream . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3-1 Example of a symmetric two-stage dispersion map with a path-average

anomalous dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3-2 Numerical simulation of DM soliton in one unit cell of simple two-stage

dispersion m ap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

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. . . . . . . . . . . . 45

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 am plifier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4-2 Experimental setup for dispersion measurements of fiber sections. . . 52

4-3 Group delay measurements (top) and dispersion calculation (bottom)

for first anomalous segment consisting of 50 km AllWaveTM . . . . . . 54

6

4-4 Group delay measurements (top) and dispersion calculations (bottom)

for second anomalous segment consisting of 25 km AllWaveTM 55

4-5 Group delay measurements (top) and dispersion calculations (bottom)

for third anomalous segment consisting of 10 km SMF. . . . . . . . . 55

4-6 Group delay measurements (top) and dispersion calculations (bottom)

for normal segment consisting of 15 km DCF. . . . . . . . . . . . . . 56

4-7 Group delay (top) and GVD (bottom) calculations for entire fiber loop

consisting of 75 km AllWaveT M , 10 km SMF, and 15 km DCF. Net

zero dispersion is achieved at 1550 nm as designed. . . . . . . . . . . 56

4-8 Experimental setup for EDFA transfer function measurements with

amplifier pump level varied between 1 and 10. . . . . . . . . . . . . . 58

4-9 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 mea-

sured 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 com-

pensation 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 mea-

sured at the loop tap point . . . . . . . . . . . . . . . . . . . . . . . . 72

A-1 Simulation of the linear case with only dispersion and no nonlinearity

or loss........ ............... ........... ........ .81

7

A-2 Simulation of fiber loop with enough amplifier gain to compensate fiber

lo ss. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2

A-3 Simulation of fiber loop with a higher pump level to show that nonlin-

earity can mitigate the shifting of the minimum pulse width position. 82

8

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 tem-

poral 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 long-

distance 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 distor-

tion 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 dis-

persion (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 dis-

advantage 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 com-

mercial 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) soli-

tons 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 in-

teraction, 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 propagat-

ing 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 disper-

sion-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 Intrinsic Fiber Properties

2.1.1 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 1O(wo) = 0 + 0 1(W - wO) + -0 2(W - W0 )2 + -0 3 (W - WO) 3 + ... , (2.1)

13

where

#4 = ., (2.2)

The first two terms, #3 wo/v, and i1 = 1/v, in Eq. (2.1), are related to the phase

(vp) and group (v_) velocities, respectively, and the third term #2 represents group-

velocity dispersion (GVD) or chromatic dispersion. GVD is the phenomenon of differ-

ent 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 fre-

quency. 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 1 27rc(23D #2 .(2.3)dA o( v9 A213

The minus sign in the equation comes from the inverse relationship between wave-

length 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. A

14

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 wave-

length) and creates an asymmetry in the dispersion profile. Polarization-mode dis-

persion (PMD) results from random polarization effects since fiber is a birefrigent

material. To first order, two eigenstates exist where the fast principal axis is or-

thogonal 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 dif-

ferent 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 + n2JE12 , (2.4)

where no is the bulk material index and n2 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,

PL =OX('E and PNL =EoX(2) 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)12E(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 (2.11)

17

and

n2 = .X (2.12)8no

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 (2.13)dt

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 n2 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 propa-

gate 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 scat-

tering, 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 commer-

cially 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 x E = (2.15)at

B9DV x H = , (2.16)at

19

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

E E _2E OPNLV x V x E+ ou + tPoEt2 - - a " t2 (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

a a2 a2PNL( __E )E = /a 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

{02O2

+ 2ik - k 2)

- E(z, t) ei(kz-wt)

We can simplify Eq.

/[t0,

2 a1982

- 2iw)

- 22wt

- pOE (at2 - 2iw aat- W2) . PPNL(Z, t)ei(kpz-wt) .(2.22)

(2.22): using the dispersion relation k2 = 1u0Cw2 and applying

the slowly varying envelope approximation. This approximation states that many

optical cycles are contained under the pulse envelope, that is,

OE a 2 Ek >

Oz az2OE a2Etat at2OPNL

W2 PNL at

(2.23)

(2.24)

(2.25)> 2 PNLat 2

Eq. (2.22) now simplifies to

2 k+ POE-- E(z, t) = 2 ( - )PNL (Z, t)ei(kp-k)z

Recall that w/k = c/n where n is the index of refraction so that poE = (n/c)2 and

Eq. (2.26) becomes

OE(z,t) poo-c nEz____+ E(z, t) + n E(zt)

Oz 2n c Ot2i(oWC( - i)PNL (Z, t)e(kp k)z

2n

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)z2n

(2.28)

Some general remarks:

1.) (kp - k) describes the difference in phase between the electric field

and the nonlinear polarization.

21

(az (2.26)

(2.27)

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 interac-

tions with one another and regain their asymptotic shapes, magnitude, and speeds.

Hasegawa and Tappert theoretically showed in 1973 that an optical pulse in a di-

electric 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 prop-

agation 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

regular solitons.

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 . NLOWC i(kp -k)z

Z =Z2n

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|2E 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 . (2.32)az C at 2cn

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

Ok1 102 kk(w) ~ k(wo) + (W - wo) + 2 (W - O) . (2.33)

Ow W 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

OFAz = (ik(w) - ik(wo))EAz , (2.35)

Oz

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 92 k= (W - WO) + (W _ wo)2 E . (2.36)

Recall from Fourier transform theory that multiplication by -i(W - wo) in the fre-

quency domain translates to a temporal derivative in the time domain. Using this

property, Eq. (2.36) becomes

OE Ok OE 1 2k O2EZ Ow WO -t 2 . (2.37)

Define 1/v 9, Ok(wo)/Ow and k" = O2k(wo)/0w 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 2E . (2.38)Oz V9 Ot 2 Ot2 2cn

Note that k is generally assumed to be equivalent to # by convention, that is, k(w)

#(w) 2. 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 (weare considering only unidirectional propagation) and the transversal component is very small for thefundamental 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 -+ t - (9'__ )z = t - /3#z and Eq. (2.38) becomes

.u(z, t) /3" 02u(z, t)2 z 2 at2 - 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) 2dxdyf f jf(x, y) 4dxdy '

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 a .#2 a 2 u /3 13U-u +2i- - -5az 2 2 t2 6 t3 + ... [higher-order dispersion terms]

SZ uU2 + 2 a(-uI2 u) -TRUY RU (2.41)1 WOa t

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 higher-

order 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]:

t - #2AWOZU(z, t) = Ao sech ) exp (-iAwot)

x exp [i ( + 12Aw2) z] exp (iq) , (2.42)

with the constraint that the parameters T and AO obey

1 -IAoI 2 (2.43)

r2 02

In Eq. (2.42), AO is the amplitude, T 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 breath-

ing (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 nonlin-

earity 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

GVDt

Iu(z,t)I Aw = d t

SPM BLUE

RED

Figure 2-1: Diagram of anomalous group velocity dispersion and Kerr nonlinearity

(self-phase modulation). Proper balancing between these two effects induces pulseshape 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 behav-

ior. 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 (anoma-

lous) 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 sys-

tems 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 previ-

ously 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 impair-

ment caused by spontaneous noise in erbium-doped fiber amplifiers (EDFA). Since

fiber is intrinsically lossy, long-distance pulse propagation requires signal amplifica-

tion 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) 1.76hwOynsp I A 2 |(G - 1)z 3 ESO1 (2.45)9 TFWHMLa E

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 dis-

placed 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 require-

ment 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 n2 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 tech-

niques 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 givendata 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 disper-

sion management [16]. Dispersion management is a technique that uses varying dis-

persions, 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 dis-

persion 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 CellNorm

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]

- _ #"L , +/ La (3 1)ave - L, + La '

#" and Lj refer to the jth segment's dispersion and length, respectively. For consis-

tency throughout this section, parameters with the subscript 'n' correspond to the

normal dispersion fiber and those with the subscript 'a' correspond to the anoma-

lous dispersion fiber. Another characteristic of a dispersion map is the map strength,

defined as

S_ | (#'Ln - 0"La)| (3.2)(32FWHM

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 mea-

sures 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-stagedispersion 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 tech-

nique) [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

u=(D+N)u (3.3)

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 = -_ a- (3.4)2 at 2 '

N =_ Z'yn2 . (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 simul-

taneously 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 disper-

sion 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, we have

(9u #2 U2 ~U./3T a~Z =t2 - Z -> = - 2,U U (3.7)az 2 t 2 2 dz 2 2

36

where U(z, w) denotes the Fourier transform of u(z, t). The solution to Eq. (3.7) is

U = Uoex ()p U) , (3.8)(-2 2

where Uo 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 two-

step 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 an-

alytic 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 dis-

persion, 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 essen-

tial 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 dispersion-

managed 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* &Bu\/ 2 B 2L = - ( U - u*aU + -2 aUa .I1 (3.12)

2 az 19z 2 at

39

Using calculus of variations, the NLSE is obtained via the variational principle

,u*, au, a*, a dzdt = 0 , (3.13)J Z *z at at

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 aL 0 aL a aL=0 -+=0 .(3.14)6u 19U az a () at a g

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 , (3.16)

40

where

(L) = f LGdt

ia A dA* - A*dA) + A2a - a3|A12 b2 + + v/2 -yaIA|4j

(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]:

6(L) 0 d d A* db6AL) = 0 - (-iaA*) = -i'a dz + A*as6A dz dz dz ± +2'IIIA (.8

- a A* 4b2+ -)+227ya|A12A ,(3.18)2 a4

6(L) d dA 3db-A* = -(iaA)=-iaz-+ Aaz6A* dz dz dz

-2 3aA 4b2 + + 2xyaIA|2 A , (3.19)-2 a a)+2/-aA2

6(L) / dA* dA\ 12a2db

6a = 0 - i A dz -A* dz + 3A2a2

- 3/ 2a2 b2 + - + V-yIA 14 = 0 , (3.20)2 a2

6(L) = 0 - d (a3 |A 12 ) = -4 2ba 3 A 12 . (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 , (3.22)dz

i A* - A dz A12 a2 a 2 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# 2ab . (3.25)dz

By comparing Eqs. (3.20) and (3.23), we obtain

db /3'A 2 =o |A3.26aT - 2# 2ab2 + 2 0 ,(3.26)

dz 20 -,F a

which can be combined with the derivative form of Eq. (3.25) to result in

-23 A2l2 (3.27)

The second-order differential equation in Eq. (3.27) yields an equation for the varia-

tion 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 fre-

quency 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 dis-

persion 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 anoma-

lous 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.

(U

z

120

100

80

60

40

20

00.2 0.4 0.6

Ak" (ps 2/km)0.8 1

Figure 3-3: Variational plot of DM soliton energy versus net normal dispersion withpulse width as a parameter (provided by Prof. Haus). The circles on the plot representdirect 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

FWHM=11ps16psU

18p

-0 - L=100kk"=17ps 2/kI

ji

. I I I-

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 ) 27Eef = (Q2 ) (/") 3 Qb [1 - (1 + 4c 2 -1/ 2 ] , (3.31)

where E refers to the pulse energy, Qb 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

true even if the average dispersion ((0") ,) is normal (positive #" /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 path-

average 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 op-

tical 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 soli-

46

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 disper-

sion 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 disper-

sion 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 tech-

niques. While numerical investigation on DM solitons is usually limited to the ideal

lossless case (especially since the variational method assumes that energy is invari-

ant), 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 exper-

imentally 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 de-

scribe 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 limi-

tations, 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 parame-

ters 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 every-

time. 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 estab-

lish 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 be-

fore 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 inter-

pretation are reported.

49

4.2.1 Laser Source

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 continuous-

wave (CW) generator such that the pulse separation is 100 ps.

To estimate the chirp in the pulses produced from the laser source, the time-

bandwidth product is calculated to determine whether it is approximately transform-

limited. 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 Pulse4500 -10

4000--20 - 4

3500 - FWHM =3000- FWHM = -30 - 1.3 nm

S 3.5 ps _02500-2000 -

- 1500 - -50 -

1000-60

500-

0 -7C-10 -5 0 5 10 1520 1530 1540 1550 1560 1570 1580 1590

Time [ps] Wavelength (nmj

Figure 4-1: Autocorrelation (left) and optical spectrum (right) of the transmitterlaser pulse. The autocorrelation FWHM translates to At = 2.5 ps and the bandwidthtranlates to Av = 160 GHz for a time-bandwidth product of about 0.4, which is closeto the Gaussian transform-limited state. The pedestal in the spectrum is ASE noiseproduced 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 characteri-

zation 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 semi-

conductor laser serving as a broadband source is modulated with an electro-optic

(EO) modulator (a Mach-Zehnder interferometer). A signal generator sets the mod-

ulation 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 ampli-

fier. An OSA can also be used to ensure that the operating wavelength lies within the

51

Tunable ~ PC MVZM PM

Laser RF

SIGNAL BIAS FIBER EDFATO BE

Signal TESTED BPF

PC - Polarization ControllerMZM - Mach Zehnder Modulator PMPM - Power Monitor (inline) TRIGGERBPF - BandPass FilterPD - PhotoDetector O t,

Sampling 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 rela-

tive 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 tech-

nique 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 1555 1560Wavelength [nm]

_,50

900 -.. .. .. -.. ..-.CL

8 5 0 - -. -. . -.... .. .. ... .. ... .... .. .

8001530 1535 1540 1545 1550 1555 1560

Wavelength [nm]

Figure 4-3: Group delay measurements (top) and dispersion calculation (bottom) foruTM

first anomalo~~~us)emn ossigo 0k l~v

54

1535 1540 1545 1550Wavelength [nm]

1535 1540 1545 1550Wavelength [nm]

1555 1560

1555 1560

Figure 4-4: Group delay measurements (top) and dispersion calculations (bottom)for second anomalous segment consisting of 25 km AllWaveTM.

1535 1540 1545 1550 1555Wavelength [nm]

1535 1540 1545 1550Wavelength [nm]

1555 1560

Figure 4-5: Group delay measurements (top) and dispersion calculations (bottom)for third anomalous segment consisting of 10 km SMF.

55

150001O

CO 10000

0.

0

U)

CU

ci:

0

-50001530

480

460

440

420

T-C

0.

CL0

0

40

1530

-8000

6000a)-a

4000

a)2000

-)U) (

1530 1560

180EC

1700

M160

0

15015 30

.. .....-. ....-. .....-. ...-. -. ..-

-..-.-.-.-.-.-

I

I -- , --- I I I I

...... .... .... . ..

.................. .......

I

5000

-.. .... -.. ... -.. ..........................

0

- - - - - - - - - - -- - - -.. ...................

- . . .. .-. .....-. . . . . . . . .

-. ...- - -. . ---. . -. . -. -.. .....-

I

I

1535 1540 1545 1550Wavelength [nm]

1535 1540 1545 1550Wavelength [nm]

Figure 4-6: Group delay measurements (top) and dispersion calculations (bottom)for normal segment consisting of 15 km DCF.

1535 1540 1545 1550Wavelength [nm]

1555 1560

50

25

0

-25

1530 1535 1540 1545 1550Wavelength [nm]

1555 1560

Figure 4-7: Group delay (top) and GVD (bottom) calculations for entire fiber loopconsisting of 75 km AllWaveTM , 10 km SMF, and 15 km DCF. Net zero dispersionis achieved at 1550 nm as designed.

56

200000n

CO

CL= -200000)

>-40000

-60000'15 30

30

CL

C,

C

0.

-1400

-1450

-1500

-1550'15

1555 1560

1555 1560

.- 4800CO,

'CD 4600

0)

cl 4400

cc1530

C

C0

C,)

CO,*0

i I -- , - - j I ---

-. - -..... ...... ..... ...

--- - - - - -

- - -

I

-. .......... -.. . -.. ..... ........ ....

-..... -. ..... -. ........ -. ..

42I

-. ...... -.. . ... ... ... .. . ... .. ..

-.. ..... .... . -

- -. - -. . .......

The dispersion profile shown in Fig. (4-7) remains fixed throughout all the ex-

periments. The path-average zero dispersion wavelength for this fiber loop is approx-

imately 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 trans-

mitter 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 Loss (dB)Anomalous 1 9.3Anomalous 2 5.0Anomalous 3 2.5

Normal 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 reflectome-

ter (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).

Fiber Section Length (km)Anomalous 1 50.1Anomalous 2 25.0Anomalous 3 10.2

Normal 15.1

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

PowerLaser Monitor Bnps oe

LIser --- with ---- >EDFA BFidpes M onitrAttenuator Fle oio

Figure 4-8: Experimental setup for EDFA transfer function measurements with am-plifier 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 levelI I I I I I

x Pump Level 1.0+ Pump Level 2.0* Pump Level 5.0o Pump Level 8.0o Pump Level 10.0

-35 -30 -25 -20 -15Input average power [dBm

-10 -5 0

Figure 4-9: Measured amplifier gain and saturation power at different pump levels.

59

15

10

5

0

m -5

0

CO

.-150

-20

-25

-30

-35-4

I0

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

OPTICALSWITCH OPTICAL

LASER SWITCH -SOURCE 50/50 COUPLER

I RF

BIASGATED

__ - TRAIN

SWITCH OPTICAL OFEDRIVERS SWITCH 50 km

ALL-WAVE

A/B A/B 35 km (RSF

ALL-WAVE

DELAY/PULSE OR SMF)GENERATOR

15 km

C/D EDFA DCF EDFA

Figure 4-10: Experimental setup of recirculating fiber loop

compensate for fiber loss with spacing arranged such that both EDFA's provide ap-

proximately 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

35 km 50 kmANOM ANOM

9 km6 km NORMNORM

Figure 4-11: Dispersion map with a tap at launch point located 7.5 km before theanomalous 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 ana-

lyzing 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 syn-

chronization 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 establishingstable 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 (usu-

ally 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 gen-

erally 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. (4-

64

Pulse width measurements at launch point50

- a, LOW power

45- -4- HIGH power

640-

35 -

30 -

'I)'

(,I'

All

025-

0

C

20 2 020 A0

05 10 15 20 25 30Number of periods around loop

Figure 4-12: Experimental data of pulse width evolution versus loop period measuredat 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. A1st

LAUNCH

35 km 50 kmnANOM ANOM

9 km6km NORM

NORM

Figure 4-13: Illustration of how the tranform-limited state position shifts in the linearcase 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 autocorrela-

tion 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 -22 ps

30- 21 psA*i-,3000 - * 4- 4-~2000 -

-2000

1000-1000

0 0-50 -40 -30 -20 -10 0 10 20 30 40 50 -50 -40 -30 -20 -10 0 10 20 30 40 50

Time [ps] Time [ps]

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 ef-

fectively "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 half-

cell 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 kmANOM

50 km 6 kmANOM NORM

9 kmNORM

Figure 4-15: Dispersion map with launch point located 7.5 km after the anomaloushalf-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 au-

tocorrelation 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 min-

imum 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

30

28

~24U_

-522

75200.C

00160

12

10

Pulse width measurements off launch pointSI I I I I I|

a - LOW power-e- HIGH power

'I

',

I I I II I 13

14 16 18 20

data of pulse width evolution versus loop period measuredFigure 4-16: Experimentalat the loop tap point

width starts to broaden as the minimum pulse width position walks away. The aber-

rant 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

2 4 6 8 10 12Nth period in loop

I

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 dispersion-

induced 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 steady-

state 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 lower-

74

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 broaden-

ing. 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 non-

76

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 en-

ergies 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 main-

tain 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 disper-

sion, 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 po-

tentially 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, col-

lisions/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 prop-

agation 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 ini-

tially done to determine whether periodically stationary pulses exist. Preliminary

numerical simulations assume lossless fiber and omit lumped amplifier gains. In addi-

tion, 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 simu-

lation, 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 ac-

count 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.

5 10 15Number of loop periods

Figure A-1: Simulation of the linear case with only dispersionloss.

and no nonlinearity or

81

- Launch.Tap Later

- - Tap Earlier

- -- -

45

40

35

30

C2

c0-25

15

10

0 20 25 30

5 10 15Number of loop periods

20 25 30

Figure A-2: Simulation of fiber loop with enough amplifier gain to compensate fiberloss.

W

-C

a,M)

0 5 10 15 20 25 30Number of loop periods

Figure A-3: Simulation of fiber loop with a higher pump level to show that nonlin-earity can mitigate the shifting of the minimum pulse width position.

82

- Launch.Tap Later

- - Tap Earlier22-

20

18

16

~14-3

12

U110

8

6

40

- Launch-.-.- Tap Later

12 - - - Tap Earlier -

11 -

10-

9

8-

7-

6

9A

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