Numerical Analysis of Nonlinear Pulse Propagation in Optical Fiber

NUMERICAL ANALYSIS OF NONLINEAR PULSE

PROPAGATION IN OPTICAL FIBER

A Thesis Presented

By

SADEED M. ZAHIN #052-652 -045

M IQBAL MAZID # 052-401-045

M. A. A. WAHID RAHAT #052-339-045

Submitted to

Department of Electrical Engineering & Computer Science in partial fulfillment of the requirements for the degree

of

Bachelor of Science in Electronics and Telecommunication Engineering (ETE)

Department of Electrical Engineering & Computer Science

North South University

Dhaka, Bangladesh.

Date: 30-12-09.

NUMERICAL ANALYSIS OF NONLINEAR PULSE

PROPAGATION IN OPTICAL FIBER

Advisor:

………………………………

Mr. Zasim U. Mazumder

Lecturer



Dhaka

Chairman:

………………………………

Dr. Miftahur Rahman

Chairman & Professor



Dhaka

i

Abstract

In this work we study the nonlinear pulse propagation in optical fiber. Dispersive effects

during propagation causes temporal spreading of pulses and it can be compensated by nonlinear

effects. When the effects are combined they can generate stable, undistorted pulses over long

distances. Numerical analysis is carried out using the split step Fourier method on different types

of pulses including hyperbolic secant, Gaussian and super-gaussian pulses of various orders.

Higher order solitons are also included in this study. Hyperbolic secant pulses are found to

propagate with constant pulse broadening ratio and linear phase change as expected. Pulse

broadening ratio of second order solitons are found to fluctuate periodically but the other higher

order solitons disintegrate as they propagate. Gaussian pulse broadening ratio appears to

decrease and the pulse stabilizes gradually to a hyperbolic secant shape with linear phase change

in the case of zero chirp parameter. Pulse broadening ratio for super-Gaussian pulses decreases

gradually.

ii

Acknowledgments

First of all we would like to thank our supervisor, Mr. Zasim U. Mozumder for

introducing us to the amazingly interesting world of Nonlinear Fiber Optics and teaching us how

to perform research work. Without his continuous supervision, guidance and valuable advice, it

would have been impossible to complete the thesis. We are especially grateful to him for

allowing us greater freedom in choosing the topic to work on, for his encouragement at times of

disappointment and for his patience with our wildly sporadic work habits. We are grateful to all

other friends for their continuous encouragement and for helping us in thesis writing. We would

like to express our gratitude to all our teachers specially Mr. Arshad Momen. Their motivation

and encouragement in addition to the education they provided meant a lot to us. Last but not

least, we are grateful to our parents and to our families for their patience, interest, and support

during our studies.

iii

To

The Chairman,

Department of Electrical Engineering & Computer Science,

North South University,

Dhaka-1217,

Bangladesh.

Summer 2009

Date: 30th

December, 2009.

Subject: Approval for valid submission of project report for the course ETE499

Dear Sir,

We, M IQBAL MAZID, SADEED M. ZAHIN and M. A. A. WAHID RAHAT are doing our final year

project/thesis, ETE 499, this semester. Our topic is “NUMERICAL ANALYSIS OF NONLINEAR PULSE

PROPAGATION IN OPTICAL FIBER”.

. We declare that the work presented in this thesis entitled “NUMERICAL ANALYSIS OF NONLINEAR

PULSE PROPAGATION IN OPTICAL FIBER”is the outcome of the investigation carried out by us under the

supervision of Mr. Zasim U. Mozumder, Assistant Professor, Department of Electrical Engineering & Computer

Science, North South University Dhaka. It is also declared that neither this thesis nor any part thereof has been

submitted or is being currently submitted anywhere else for the award of any degree or diploma. We have completed

our project and will be doing our presentation on 30th

December, 2009.

Under light of this letter, we hope that our report is granted for approval and is adjudged a valid submission in

fulfillment of the course ETE-499.

Sincerely,

M IQBAL MAZID

Student ID.: 052-401-045

………………………………

SADEED M. ZAHIN

Student ID.: 052-652 -045

………………………………….

M. A. A. WAHID RAHAT

Student ID.: 052-339-045

………………………………….

iv

TABLE OF CONTENTS

1. INTRODUCTION 1

1.1 INTRODUCTION 2

1.2 THESIS OBJECTIVE 3

1.3 BACKGROUND OF THE PROJECT 4

2. LITERATURE REVIEW 7

2.1 DISPERSION 8

2.1.1 GROUP VELOCITY AND PHASE VELOCITY 9

2.1.2 GROUP VELOCITY DIPERSION 9

2.1.3 THIRD ORDER DISPERSION 10

2.2 NONLINEAR EFFECTS 11

2.2.1 NONLINEAR REFRACTIVE INDEX 11

2.2.2 TYPES OF NONLINEAR EFFECTS 12

2.2.2.1 SELF-PHASE MODULATION 12

2.2.2.2 STIMULATED RAMAN SCATTERING 13

2.2.2.3 STIMULATED BRILLOUIN SCATTERING 14

2.3 PULSE PROPAGATIONS 15

2.3.1 NONLINEAR SCHRODINGER EQUATION (NLSE) 16

2.3.1.1 GENERALIZED NLSE 16

2.3.1.2 SIMPLIFIED NLSE 17

2.3.1.3 NORMALIZED NLSE 18

v

2.3.2 TYPES OF PULSES 19

2.3.3 HYPERBOLIC SECANT PULSES 19

2.3.4 GAUSSIAN PULSES 19

2.3.4.1 CHIRPED GAUSSIAN PULSE 21

2.3.5 SUPER-GAUSSIAN PULSES 22

2.4 CHIRP 23

2.4.1 CHIRP DEFINATION 23

2.4.2 TYPES OF CHIRP 24

2.4.2.1 GVD INDUCED CHIRP 25

2.4.2.2 SPM INDUCED CHIRP 25

2.4.2.3 PRE INDUCED CHIRP 25

2.4.3 CRITICAL CHIRP 25

2.4.4 EFFECTS OF CHIRP 26

2.5 SOLITON 27

2.5.1 CONDTIONS FOR SOLITON 27

2.5.2 HIGHER ORDER SOLITON 27

3. DESCRIPTION OF METHOD 28

3.1 SPLIT STEP FOURIER METHOD 29

3.2 SYMMETRIZED SPLIT STEP FOURIER METHOD 32

3.3 DECISION 34

4. IMPLEMENTATION 35

4.1 NONLINEAR SCHRODINGER EQUATION (NLSE) SOLUTIONS 36

4.1.1 GENERALIZED NLSE 36

4.1.2 SIMPLIFIED NLSE 36

4.1.3 NORMALIZED NLSE 37

vi

4.2 GAUSSIAN IMPLEMENTATION 38

4.3 SOLITON IMPLEMENTATION 40

4.4 FLOW CHART 42

4.5 PULSE BROADENING RATIO 45

4.5.1 FULL WIDTH HALF MAXIMUM 45

5. ANALYSIS OF RESULTS 46

5.1 ANALYSIS PLAN 47

5.2 GAUSSIAN PULSE ANALYSIS 48

5.2.1 ANALYSIS FOR GVD VARIATION 49

5.2.2 ANALYSIS FOR NONLINEAR PARAMETER VARIATION 51

5.2.3 ANALYSIS FOR INPUT POWER VARIATION 53

5.3 SOLITON PULSE ANALYSIS 55

5.4 HIGHER ORDER SOLITON ANALYSIS 58

5.5 CHIRPING EFFECT ANALYSIS 63

5.5.1 Analysis for Un-chirped Gaussian pulse 64

5.5.2 Analysis for Gaussian pulse of Chirp = -1 66

5.5.3 Analysis for Gaussian pulse of Chirp = -2 68

5.5.4 Analysis for Gaussian pulse of Chirp = 2 70

5.6 SUPER GAUSSIAN PULSE ANALYSIS 72

5.6.1 Analysis for Supergaussian pulse for chirp =0 75

5.6.2 Analysis for Supergaussian pulse for chirp =100 76

6. CONCLUSION 77

BIBLOGRAPHY 79

MATLAB CODES 81

LIST OF SYMBOLS 92

vii

List of Figures and Tables

Fig 2.1: Group Velocity (red) and Phase Velocity (blue).

Fig 2.2: Elastic Rayleigh scattering and Inelastic Raman Scattering

Fig 2.3: Raman Scattering

Fig2.4: Chirp

Fig 3.1: Split step Fourier method

Fig 3.2: Symmetrized split step Fourier method

Fig 4.1: Gaussian Flowchart

Fig 4.2: Soliton Flowchart

Fig 4.3: Supergaussian Flowchart

Fig 4.4: Full Width at Half Maximum

Fig5.1: Input Gaussian Pulse

Fig5.2: Pulse Broadening Ratio for different values of GVD

Fig 5.3: Pulse Broadening Ratio for different values of Nonlinearity

Fig5.4: Pulse Broadening Ratio for different values of Power

Fig5.5: Input Hyperbolic secant Pulse

Fig5.6: Soliton Pulse Propagation

Fig5.7: Pulse Broadening Ratio for Soliton Pulse Propagation

Fig5.8: Pulse propagation for higher order soliton for N=2.


viii

Fig5.10: Pulse propagation for higher order soliton for N=5

Fig5.11: Pulse Broadening Ratio for Higher Order Soliton with different N

Fig5.12: Pulse Broadening Ratio of Gaussian Pulse with different chirp

Fig5.13: Pulse Propagation for chirp 0

Fig5.14: Pulse Propagation for chirp -1



Fig5.17: Input pulse for different values of m.

Fig5.18: Supergaussian pulse propagation for m=4

Fig5.19 Supergaussian pulse propagation for m=10



Fig5.22: Pulse Broadening Ratio for different values of m for chirp 0

Fig5.23: Pulse Broadening Ratio for different values of chirp for m=100

Table 4.1: Gaussian input

Table 4.2: Super-Gaussian input

Table 4.3: Hyperbolic Secant input

Table 5.1: Test Plan

CHAPTER 1: INTRODUCTION

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CHAPTER 1

INTRODUCTION


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1.1 INTRODUCTION

In this thesis numerical analysis of nonlinear pulse propagation is carried out. This is done

mainly by solving the nonlinear Schrodinger equation using the symmetrized split step fourier

method. In a nonlinear media, dispersive effects exist simultaneously with nonlinear effects.

Refractive index dependence on intensity results in optical Kerr effect which causes narrowing

of transmitted pulses by inducing self phase modulation while second order group velocity

dispersion causes the pulses to spread. In this dissertation, group velocity dispersion is discussed

in relative detail followed by self phase modulation. These individually detrimental effects are

shown to combine beneficially for propagation of pulses here. The importance of chirp is

discussed along with its various properties. Induced chirp and input chirp are differentiated and

explained using equations.

Gaussian, super-gaussian pulses and hyperbolic secant pulses are studied and propagated by

using them as input in to the nonlinear Schrodinger equation. The symmetrized split step fourier

method is described in depth. Explanation of each step is included along with the relevant

equations defining these steps. A descriptive method of how symmetrized split step fourier was

incorporated as a solution of both simplified and normalized nonlinear Schrodinger equations is

presented in the implementation section. Pulse evolutions of these pulses under varying

circumstances and parameters are presented in this body. The purpose of obtaining numerical

calculations where done by calculating the pulse broadening ratio of the various pulses

propagated under different conditions [1]. Width used is a measure of the full width half

maximum in all cases.


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1.2 THESIS OBJECTIVE

In this thesis, we have primarily studied the effects of non-linear pulse propagations which

include hyperbolic secant, Gaussian, Super Gaussian pulses. We shown certain effects of chirp

on these pulses and using pulse broadening (PB) ratio described their effects. We also included

Super Gaussian pulses of higher order in this thesis and shown PB ratios of higher order soliton.


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1.3 BACKGROUND OF THE PROJECT

From the very beginning of the human race, people wanted to convey information quickly as

well as precisely. In order to do so, they tried to transfer information using different methods and

medium. Since the evolution of modern science, there were more efficient ways to transfer

information. Recently optical communication system has been the groundbreaking development.

Optical communication system was first conceived in the late 18th century by a French Engineer

Claude Chappe who created an optical telegraph. Since the advent of telegraphy in the 1830s,

electrical communication systems became the first foremost modern communication method.

Until the early 1980s, most of fixed (non-radio) signal transmission was carried by metallic cable

(twisted wire pairs and coaxial cable) systems. However, large attenuation and limited bandwidth

of coaxial cable limited its capacity upgrade. The bit rate of the most advanced coaxial system

which was used in the United States in 1975 was 274 Mb/s [1]. At approximately the same time,

there was a need of switching from analogue to digital transmission to improve transmission

quality, which calls for further boost of transmission bandwidth. Many efforts were prepared to

surmount the limitations of coaxial cable during the 1960s and 1970s. In 1966, Kao and

Hockham proposed the use of optical fiber as a guiding medium for the optical signal [1]. Four

years later, a major innovation occurred when the fiber loss was reduced to about 20dB/km from

previous values of more than 1000dB/km. From that time, optical communication technology has

developed rapidly to achieve larger transmission capacity and longer transmission distance. As a

result the transmission capacity has been increased about 100 fold in every 10 years [1].

The first generation of optical communication was designed with multi-mode fibers and direct

bandgap GaAs light emitting diodes (LEDs) which operate at the 0.8nm- 0.9nm wavelength

range [18]. In contrast to the typical repeater spacing of coaxial system (~1km), the longer

repeater spacing (~10km) was a major motivation. Large modal dispersion of multi-mode fibers

and high fiber loss at 0.8nm (> 5dB/km) limited both the transmission distance and bit rate. In

the second generation, multi-mode fibers were replaced by single-mode fibers, and the center


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wavelength of light sources was shifted to1.3nm, where optical fibers have minimum dispersion

and lower loss of about 0.5 dB/km.

However, there was still a strong demand to increase repeater spacing further, which could be

achieved by operating at 1.55nm where optical fibers have an intrinsic minimum loss around

0.2dB/km. Larger dispersion in the 1.55nm window delayed moving to a new generation until

dispersion shifted fiber became available. Dispersion shifted fibers reduce the large amount of

dispersion in the1.55nm window by modifying the index profile of the fibers while keeping the

benefit of low loss at the 1.55nm window [1].

However, growing communication traffic and demand for larger bandwidth per user revealed a

significant drawback of electronic regenerator systems, namely inflexibility to upgrade. Because

all the regenerators are designed to operate at a specific data rate and modulation format, all of

them needed to be replaced to convert to a higher data rate. The difficulty of upgradeability has

finally been removed by optical amplifiers, which led to a completely new generation of optical

communication. An important advance was that an erbium-doped single mode fiber amplifier

(EDFA) at 1.55nm was found to be ideally suited as an amplifying medium for modern fiber

optic communication systems [18]. Invention of the EDFA had a profound impact especially on

the design of long-haul undersea systems.

It is highly likely that WDM systems will bring another big leap of transmission capacity of

optical communication systems. Some research groups have already demonstrated that it is

possible to transmit almost a Tbits/s of total bit rate over thousands of kilometers. In 1999, for

example, N. Bergano et al. successfully demonstrated transmission of 640 Gb/s over 7200km

using a re-circulating loop while G. Vareille et al. demonstrated the transmission capacity of

340Gb/s over 6380km on a straight-line test bed [1]. These results indeed show that remarkable

achievements have been made in recent years, and let us forecast that optical communication

systems in the next generation will have a transmission capacity of a few hundreds of Gb/s.


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While high capacity dense WDM systems keep heading to closer channel spacing and broader

bandwidth of optical amplifiers to fully exploit the fiber bandwidth, on the other hand, upgrading

embedded systems remains as another challenge. As of the end of 1997, about 171 million km of

fiber have been deployed worldwide, of which 69 million km is deployed in North America [1].

Unfortunately, most of the embedded fibers are conventional single-mode fibers which have a

large dispersion at the 1.55nm window. Upgrading these systems will require various dispersion

combating techniques which are highly tuned at a specific system to optimize system

performance.

CHAPTER 2: LITERATURE REVIEW

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CHAPTER 2

LITERATURE REVIEW


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2.1: DISPERSION

Dispersion is an impediment in optical fiber transmission, especially

in case of single mode fiber. Even though the word “Chromatic” refers to color of the light

pulse, it is the wavelength of a light pulse that is more relate to this type of dispersion.

Chromatic dispersion appears in optical fiber when an incoming light pulses turn up at the

receiver site at different time instants even if the pulses travel same path along the fiber. It

may appear that the chromatic dispersion is predominant fact in multimode fiber but it is

not that much of importance as modal dispersion is the dominant factor in multimode

dispersion [2]. Now if we concentrate on single mode fiber, it is indeed a dominant factor.

One query may arise in the process that if there is only one mode in single mode fiber; then

how come chromatic dispersion is a predominant limiting factor as it is consist of material

dispersion and waveguide dispersion. In single mode fiber, we may encounter the problem

as a mode might be composed of light pulses of different colors (different wavelengths).

Pulse broadening as a result of chromatic dispersion can be evaluated as follows:

LDtchrom )(

Here, D ( ) =chromatic dispersion parameter measured in picoseconds (ps) per nanometer

(nm) and kilometer (km) and spectral width of a light source in nm.

If we consider the above equation once again, then we see that dispersion parameter can be

positive or negative. When D is less than zero, then GVD parameter 2 is positive. Hence the

dispersion is positive and is known as normal dispersion region. On the contrary, if the value of

D is greater than zero, then the GVD parameter 2 is negative which is known as anomalous

dispersion region [2] [3].

In case of a single-mode optical fiber, the zero-dispersion wavelength is the wavelength for

which material dispersion and waveguide dispersion eliminates each other. In silica-based optical

fibers, minimum material dispersion occurs naturally at a wavelength of approximately 1300nm.

Again, if we consider multimode optical fiber, it is the wavelength at which the amount of

material dispersion is minimum or zero. Thus it is also known as minimum dispersion

wavelengths.


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2.1.1 PHASE VELOCITY AND GROUP VELOCITY

Phase velocity indicates the ideal characteristics of the ideal signals unlike a practical case of

combining group of signals. Phase velocity is the velocity of a certain phase point of an

Electromagnetic (EM) wave. If velocity is v, then v= / ; where and both are time

independent. Group velocity can be obtained by differentiating with respect to angular

frequency, . Group velocity can be referred as the speed at which the light power propagates

along the fiber in a specific mode. Any information signal and power travel at the group not at

the phase velocity.

Fig 2.1: Group Velocity (red) and phase Velocity (blue).

2.1.2 GROUP VELOCITY DISPERSION

Different frequency components of the light pulses traveling at different velocities produces

spreading light pulses in time and this phenomenon is known as Group Velocity Dispersion

(GVD). GVD also enumerates as the group delay dispersion parameter, D, which is applicable

for only uniform medium.

c

D

2

2

d

nd =

d

d(

gv

1) = -

22

2

c

(2.1)


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Here, 2 =

2

2

d

dis GVD parameter which determines the amount of pulse broadening on

propagation along the fiber from the above equation it is evident that GVD is also wavelength

dependent. The frequency dependence of the group velocity leads to pulse broadening simply

because different spectral components of the pulse disperse during propagation and do not arrive

simultaneously at the fiber output.

2.1.3 THIRD ORDER DISPERSION

In most cases pulse broadening due to the lowest-order Group Velocity Dispersion (GVD) term

proportional to β2. However in some cases the third-order term proportional to β3 also plays an

important role as well. If we Consider .the basic pulse propagation equation stated bellow, then

we see β2 is the dominant case in most cases.

(2.2)

However, if the light pulse wavelength is nearly similar with the zero-dispersion wavelength,

then β2 term becomes zero. As a result the β3 term then offers the principal part to the GVD

effects. For ultra short pulses (width T0 < 1 ps), it is necessary to include the β3 term even when

β2 is not equal to 0 because the expansion parameter Δω=ω0 is no longer small enough to justify

the truncation of the expansion in the following equation after the β2 term[3].

(2.3)

One practical example of third order dispersion is in case of mode-locked lasers which have

pulse durations nearly less than 30 femtoseconds [3], it is essential to supply dispersion

compensation for the group delay dispersion as well as for the third-order dispersion and

possibly for even higher orders.


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2.2 NONLINEAR EFFECTS

2.2.1 Non Linear Refractive Index

Nonlinear effects in fiber optic are mostly generated due to nonlinear refraction. Refractive index

generally defines the density of the propagating medium thereby describing how light would

propagate and at what speed. When light of very high intensity is launched into the fiber, an

additional variable contributes to the total refractive index. The total refractive index is defined

as follows:

2

20 Innn (2.4)

Here, n 0 is the material refractive index and it is assumed constant for a certain frequency. The

second term is 2n |I| 2 defined as the nonlinear refractive index where 2n is nonlinear index co-

efficient. When the light pulses are having short wavelengths and very high intensities (such as

output of a laser) may vary a refractive index of a medium which as a result may give rise to

nonlinear optics. If the refractive index of the medium varies nonlinearly with the field (linearly

with the intensity), then it is known as the optical Kerr effect [3] and results in phenomena such

as self-focusing and self-phase modulation. If the index varies linearly with the field then it is

known as the Pockels effect.

The value of n 2 is affected by the experimental technique used to measure it. The reason is that

two other mechanisms which is related to molecular motion (the Raman effect) and excitation of

acoustic waves through electrostriction (Brillouin scattering), also contribute to n 2 . However,

their relative contributions depend on whether the pulse width is longer or shorter than the

response time associated with the corresponding process. The electrostatic contribution

diminishes for pulses shorter than 100 ps but reaches its maximum value (_16% of total n 2 ) for

pulse widths >10 ns. In contrast, the Raman contribution does not vanish until pulse widths is

less than 50 fs and is close to 18% for pulse widths greater than 10ps. However, we should be

careful when we are measuring the value using different pulse widths.


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As silica fiber has relatively low value of n2

, several other kinds of glasses with larger

nonlinearities have been used to make optical fibers . For a lead-silicate fiber n 2 was measured to

be 210 19 m 2 /W. In chalcogenide As 2 S 3 -based fibers, the measured value n2

= 4.210 18 m 2

/W can be larger by more than two orders of magnitude compared with the value for silica fibers

[3]. Such fibers are attracting increasing attention for applications related to nonlinear fiber

optics in spite of their relatively high losses. Their use for making fiber gratings and nonlinear

switches has reduced power requirements considerably.

2.2.2 TYPES OF NONLINEAR EFFECTS

2.2.2.1 SELF-PHASE MODULATION

Self-phase modulation (SPM) refers to the self-induced phase shift due to an optical field during

its propagation in optical fibers. As we know the refractive index of a medium depends on the

intensity of the pulse propagating through that medium but this can be ignored most of the times

except when pulse intensity becomes large. As ultra short pulses have high intensity, thus self

phase modulation occurs when these ultra short light pulses, whose time duration is on the order

of the femtosecond (10 -15 s), are propagated through an optical fiber. When they propagate

through an optical fiber, the varying intensity across the pulse envelope will face varying

refractive index and as a result the pulse spectral components will experience varying frequency

shift. . This results in pulse broadening which causes overlapping between adjacent bits and

limits maximum bit transfer rate.

The GVD broadens optical pulses during their propagation inside an optical fiber except when

the pulse is initially chirped in the right way. More specifically, a chirped pulse can be

compressed during the early stage of propagation whenever the GVD parameter 2 and the

chirp parameter C happen to have opposite signs so that β2C is negative [2]. The nonlinear

phenomenon of SPM imposes a chirp on the optical pulse such that C > 0. Since β 2 < 0 in the

1.55- μm wavelength region, the condition β 2C < 0 is readily satisfied. Moreover, as the SPM-

induced chirp is power dependent, it is not difficult to imagine that under certain conditions,


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SPM and GVD may cooperate in such a way that the SPM-induced chirp is just right to cancel

the GVD-induced broadening of the pulse [3]. The optical pulse would then propagate

undistorted in the form of a soliton.

2.2.2.2 Stimulated Raman scattering

Raman scattering is the inelastic type of scattering which implies that the kinetic energy is not

conserved in this scattering process. When light pulse is scattered from an atom, most photons

show elastically scattered nature which means that that after scattering the photons contain the

same energy or frequency and wavelength as compared to the incident photons.

Fig 2.2: Elastic Rayleigh scattering and Inelastic Raman Scattering

This process is known as Rayleigh scattering. In case of Raman scattering, only a small fraction

of the light pulses are scattered by an excitation where the scattered photons have a different

frequency. In most cases this frequency is lower than the frequency of incident photons. It is

evident now that this scattering process results in loss of power but at low power lever, this loss

is negligible since the scattering cross sections are small enough. However, at higher power level


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this Stimulated Raman Scattering (RBS) process emerges as an important phenomenon. As the

incident power exceeds a threshold value the intensity of the scattered light raises exponentially.

Stimulated Raman Scattering can take place in both the backward direction and the forward

direction of the optical fiber [3].

Fig 2.3: Raman Scattering

2.2.2.3 Stimulated Brillouin Scattering

Stimulated Brillouin Scattering (SBS) is another instance of inelastic scattering of photons.

However, the main difference is that the optical phonons participate in Raman scattering. On the

other hand acoustic phonons participate in Brillouin scattering. At the beginning, SRS and SBS

are quite similar in nature, in case of single mode fiber different dispersion relations for acoustic

and optical phonons pinpoint following differences:

1. SBS occurs only in the backward direction; on the contrary SRS can occur in both directions.

2. The scattered light is shifted, which is known as Stokes shift, in frequency by about 10 GHz

for SBS but for SRS it is up to 13 THz.

3. The Brillouin gain spectrum possesses extremely narrow bandwidth (less than 100MHz)

compared to Raman-gain spectrum that extends over 20–30 THz [3].


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2.3 PULSE PROPAGATION

The propagation of optical fields inside fibers like all electromagnetic fields are described by

Maxwell’s equation.

t

BE

(2.5)

t

HJE

(2.6)

D (2.7)

Wave equation describes propagation of light in optical fibers. The following wave equation can

be obtained by mathematical manipulation of the Maxwell’s equations in terms of E and P ,

where P is the induced polarization of the wave [2] [12].

2

2

02

2

2

1

t

P

t

E

cE

(2.8)

Where c is the speed of light in vacuum and the relation 2

00 /1 c

When we consider the nonlinear domain, P can be dissected to obtain linear and nonlinear parts

as shown:

NLL PPP (2.9)


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LP = Linear part of induced polarization

NLP = Nonlinear part of induced polarization

Using the equation …..(2.8) and equation …….(2.9)we obtain the equation :

2

2

02

2

2

1

t

P

t

E

cE

(2.10)

This wave equation is used as the starting point for deriving the nonlinear Schrödinger equation

which will be explained in the following section.

2.3.1 NONLINEAR SCHRODINGER EQUATION (NLSE)

2.3.1.1 Generalized NLSE

)(

1

262

2

2

0

2

3

3

3

2

2

2

T

ATiAA

TAAAiA

T

A

T

Ai

z

AR

Self Steepening

Group Velocity Dispersion Nonlinearity

Attenuation Raman Scattering


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Third Order Dispersion

)(

1

262

2

2

0

2

3

3

3

2

2

2

T

ATAA

TA

iAAiA

T

A

T

Ai

z

AR

(2.11)

2.3.1.2 Simplified NLSE

AAAT

A

T

Ai

z

A 2

3

3

3

2

2

2

262

(2.12)

AAAT

Ai

z

A 2

2

2

2

22

(2.13)

Nonlinear parameter

;2 2

effA

n

(2.14)

effA effective core area of the fiber ;

2n = nonlinear index coefficient ;

= optical wavelength


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0T

T

(2.15)

DL

zZ

(2.16)

0P

AU

(2.17)

The previous normalization parameters are used in transforming the simplified NLSE to

normalized NLSE.

2.3.1.3 Normalized NLSE

AUUN

AUs

Z

Ui

262

22

3

3

3

2

2

(2.18)

02

22

2

2

UUN

Us

Z

Ui

(2.19)

02

22

2

2

UUiN

Usi

Z

U

(2.20)

s=sign( 2 )= 1

If s=1 Normal GVD


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If s=-1 Anomalous GVD

DLPN 0

2 =2

00

TP

(2.21)

2.3.2TYPES OF PULSES

In this study various type of input pulses are used to analyze their propagation through nonlinear

media. Among them are the following :

1. Hyperbolic secant pulses – equations, soliton : describe soliton etc, GVD cancels SPM

2. Gaussian pulses

3. Super-Gaussian pulses

2.3.3 HYPERBOLIC SECANT PULSES

The optical field for such pulses takes the form of:

TohNU

sec),0(

(2.22)

2

2

2expsec),0(

To

iC

TohNU

(2.23)

2.3.4 GAUSSIAN PULSE

The incident field for gaussian pulses can be written as


20 | P a g e

2

0

2

2exp*),0(

TAoU

(2.24)

0T is the initial pulse width of the pulse .

In theory, gaussian pulses while propagating maintain their fundamental shape , however their

amplitude width and phase varies over given distance [2] .

Many quantitative equations can be followed to study the properties of gaussian pulses as it

propagates over a distance of z .

Amplitude of a gaussian pulse varies with the equation

2

2

2

2/1

2

2

0

00 2exp),0(

iTiT

TTU

(2.25)

Width of a gaussian pulse varies with this equation :

2/12

01 1)(

DL

zTz

(2.26)

Under the effect of dispersion this equation shows how the gaussian pulse width broadens over z

.

Phase of the propagating pulse can be numerically determined with this formula :


21 | P a g e

D

D

D

L

z

T

L

z

L

zs

z 1

2

0

2

2

tan2

1

)(1

)(

),(

;

(2.27)

s=sign( 2 )

The phase changes with respect to time .The derivative of phase change is called chirp

which will be described later on.

2.3.4.1 CHIRPED GAUSSIAN PULSE

Gaussian pulses with initial chirp have the following input field

2

2

2

)1(exp*),0(

To

iCAoU

(2.28)

A will be shown later , the effect of initial chirp results in varied behavior of the gaussian pulse

for various chirp parameters C . The amplitude, width and phase react to the inclusion of chirp

and effects depend on the sign and magnitude of the chirp [3].

Consider a width 1T after a distance of z. The broadening of chirped gaussian pulse can be

written as

2/12

2

0

2

2

2

0

2

0

1 1

T

z

T

zC

T

T

(2.29)


22 | P a g e

The broadening depends upon the signs of 2 and C.

If 2 C<0 , pulse width becomes minimum :

2/12

0min

1)1( C

TT

(2.30)

at a distance of

DL

C

Cz

2min1

||

(2.31)

2.3.5 SUPERGAUSSIAN PULSES

The input field for such pulse is described by

2

2

2

)1(exp*),0(

To

iCAoU

m

(2.32)

Here m is the order of the supergaussian pulse and determines the sharpness of the edges of the

input . The higher the value of m steeper is the leading and trailing edges of the pulse .


23 | P a g e

As we continue increasing m , we eventually get a rectangular pulse shape which evidently has

very sharp edges. In case of m=1 we get the gaussian chirped pulse.

The sharpness of the edges plays an important part in the broadening ratio because broadening

caused by dispersion is sensitive to such a quality [12].

2.4 CHIRP

2.4.1 CHIRP DEFINITION

Chirp of an optical pulse is the change of carrier frequency with respect to time in a deterministic

fashion. If the instantaneous frequency of the pulse increases with time then it is called an up-

chirp. Similarly, if the instantaneous frequency of the pulse reduces with time then it is called a

down-chirp.

Input pulse


24 | P a g e

Fig2.4: Chirp

An input pulse with chirping can be expressed as-

(2.33)

Where, C is a chirp parameter. If a positive chirp is applied (C>0), the frequency of the pulse

will increase while propagating through the fiber. Oppositely, the frequency will decrease if

negative chirp (C<0), is applied.

We have seen before, the phase of a pulse propagating through a nonlinear medium changes

along time. Frequency of a pulse is nothing but the change of phase of that pulse. The frequency

also changes from initial frequency as a result of time dependence of phase across pulse [13].

This frequency change for a Gaussian pulse can be calculated by

(2.34)


25 | P a g e

2.4.2 Types of Chirp:

Based on induction, there are three types of Chirp.

GVD induced Chirp

SPM induced Chirp

Pre induced Chirp

2.4.2.1 GVD induced Chirp:

GVD induced Chirp occurs due to the group velocity dispersion effect while pulse propagate

through an optical fiber. The chirp is linear and negative over a large central region. It results spreading of

pulse width from the original pulse width. It is positive near the leading edge and becomes negative near

the trailing edge of the pulse [2].

2.4.2.2 SPM induced Chirp:

SPM induced chirp is up-chirp or positive-chirp that occurs due to the nonlinear self phase

modulation effect inside an optical fiber. The chirp is linear and positive over a large central region of the

Gaussian pulse. SPM induced Chirp makes the pulse width narrower from the original pulse width as

pulses propagate through an optical fiber. It is negative near the leading edge and becomes positive near

the trailing edge of the pulse.

2.4.2.3 Pre induced Chirp:

Pre induce chirp is the chirp added to an input pulse before sending it through an optical fiber. Pre

induced Chirp can be either positive or negative. Pre induce Chirp is used to balance the GVD induced

chirp and SPM induced chirp. Also chirping is added for archiving the following advantages-


26 | P a g e

Suppression of four wave mixing

Lowering pulse width fluctuations

Spectral compression

And generating transform limited output pulses

2.4.4 Critical Chirp:

Critical chirp is the maximum value of chirp of an input pulse for which soliton can be formed.

Chirp disturbs the balance between group velocity dispersion and self phase modulation.

Therefore chirp of the input pulse is kept to a minimum level for propagating soliton pulses. Furthermore,

higher chirp induced to an input pulse results loss of the pulse energy while forming the soliton as

dispersive waves. That’s why soliton cannot be formed if the chirping crosses a certain value. That value

of chirp is called Critical Chirp. For first order soliton the critical chirp value is 1.64.

2.4.5 Effects of Chirp:

GVD induced Chirp:

Hare effects of GVD induced chirp is only for anomalous dispersion (β₂ <0)

When Pre Chirp is positive(C>0) : instantaneous frequency increases linearly from the leading to

the trailing edge (up chirp)

When β₂ C<0: dispersion induced chirp(β₂) is in opposite direction to Pre Chirp (C). As a result

net chirp is reduced. Pulse width will be narrowing at first. After a distance, GVD induced chirp

dominates C and pulse spreads.

When C<0: Instantaneous frequency decreases linearly form the leading edge to the trailing edge

of the propagated pulse (up chirp).

When β₂ C>0: Pulse width broadens monotonically.

SPM induced Chirp:

SPM induced frequency chirp is linear and positive over central portion of Gaussian pulse.


27 | P a g e

Positive chirp: (C>0): The SPM induced chirp and initial pre chirp adds together. And Results

Pulse width broadening together.

Negative chirp (C<0): Deducts positive SPM induced chirp from negative pre chirp. The result

depends on the magnitude of the pre chirp. If the value of pre chirp is higher than the SPM

induced chirp, then the result will be pulse width broadening [13]. If the pre induce chirp is lower

than the SPM induced chirp, then the pulse width will be spread at first, but as propagating

distance becomes higher, SPM induced chirp becomes dominant and the pulse width narrows.

2.5 SOLITON:

Soliton in optical fiber generally means special type of pulse that can propagate undistorted over long

distance. Soliton is a result of dispersion and nonlinear effects cancelling each other. Red shift at the

leading edge of a pulse is caused by Self phase modulation [2.]In anomalous dispersion region blue shift

occurs at the leading edge of a pulse. As a result in anomalous dispersion these effects cancel each other

and the input pulse maintain its shape in both time domain and frequency domain.

2.5.1: CONDITIONS FOR SOLITON:

The conditions for soliton are

1) The dispersion region must be anomalous. That is <0.

2) The input pulse must be an un-chirped hyperbolic secant pulse. In our simulation we used the

following pulse-

TohU

sec),0(

(2.35)


28 | P a g e

3) The pulse energy has to be-

(2.36)

4) The dispersion length must be approximately the same as the nonlinear length.

5) The GVD induced chirp should exactly cancel the SPM induced chirp.

2.5.2: Higher Order Soliton:

Higher order soliton are soliton with higher energy. More specifically, the energy of a higher order soliton

is square of an integer number times higher than a fundamental soliton.

In our simulation we used –

TohNU

sec),0(

(2.37)

Higher order soliton do not have a fixed pulse shape like fundamental soliton. But they gain their shape

periodically.

CHAPTER 3: DESCRIPTION OF METHOD

Page | 28

CHAPTER 3

DESCRIPTION OF METHOD


Page | 29

3.1 SPLIT STEP FOURIER METHOD

The split step fourier method is a pseudo-spectral numerical method for solving

partial differential equations such as the nonlinear Schrödinger equation. It is

applied because of greater computation speed and increased accuracy compared to

other numerical techniques.

Dispersion and nonlinear effects act simultaneously on propagating pulses during

nonlinear pulse propagation in optical fibers. However, analytic solution cannot

be employed to solve the NLSE with both dispersive and nonlinear terms present.

Hence the numerical split step fourier method is utilized, which breaks the entire

length of the fiber into small step sizes of length ‘h’ and then solves the nonlinear

Schrödinger equation by splitting it into two halves , the linear part (dispersive

part) and the nonlinear part over z to z + h [7]. Each part is solved individually

and then combined together afterwards to obtain the aggregate output of the

traversed pulse. It solves the linear dispersive part first, in the fourier domain

using the fast fourier transforms and then inverse fourier transforms to the time

domain where it solves the equation for the nonlinear term before combining

them. The process is repeated over the entire span of the fiber to approximate

nonlinear pulse propagation. The equations describing them are offered below [7].

The value of h is chosen for ,|| 2

max hAp where rad05.0max and Ap =

peak power of A (z, t); max = maximum phase shift

In the following part the solution of the generalized Schrödinger equation is

described using this method.


Page | 30

T

ATiAA

TAAAiA

T

A

T

Ai

z

AR

2

2

0

2

3

3

3

2

2

2 )(1

262

(3.1)

ANLt

A)ˆˆ(

(3.2)

The linear part (dispersive part) and the nonlinear part are separated.

Linear part

A

T

A

T

AiL

262ˆ

3

33

2

2

2

(3.3)

Nonlinear part

T

ATiAA

TAAAiN R

2

2

0

2)(

1ˆ

(3.4)

Solution

),()]ˆˆ(exp[),( tzANLhThzA (3.5)

),()êxp()êxp(),( tzANhLhThzA (3.6)

Linear part solution

The solution of the linear part )êxp( Lh is done in the fourier domain by using the

identity iT

. Since in the fourier domain is simply a numerical sequence


Page | 31

of digits, the calculation that would otherwise be complicated in the time domain

due to computation of the differential terms are mitigated in the fourier domain.

)êxp( Lh )](êxp[ iLh and A(z,t) ),( 0 zA

Linear Solution = ),()](êxp[ 0

1

zAiLhFT

(3.7)

Nonlinear part solution

)êxp(),( NhThzA Linear Solution (3.8)

In our study as will be shown later, we have considered 3 =0, 0 and higher

order nonlinear effects of self steepening and Raman scattering (reserved for ultra

short pulses) to be 0 i.e. 0)(1

2

2

0

T

ATiAA

TAR

. We make this

assumption because the pulse width chosen is of the order of picoseconds.

Fig 3.1: Split step fourier method


Page | 32

3.2 SYMMETRIZED SPLIT STEP FOURIER METHOD

The accuracy of the SSFM can be further improved by using the symmetrized

form of the method. In this technique, the linear dispersive part is solved in two

equal steps each of size h/2. To begin with, the first half of the linear part is

solved in the fourier domain over z to z +h/2 [7].It is then inverse fourier

transformed and then combined with the nonlinear part which is solved in the

time domain using step size h. The resultant outcome is then multiplied with the

second half of the linear dispersive part over z to z +h/2 in the fourier domain.

Fourier transformation is done using the fast fourier transform method as before.

Solution

If h is small, then:

),()]ˆ2

exp()ˆexp()ˆ2

[exp(),( tzALh

NhLh

ThzA (3.9)

If h is larger than accuracy can be improved using this equation:

),()]ˆ2

exp()(ˆexp)ˆ2

[exp(),( tzALh

zdzNLh

ThzAhz

z

(3.10)

Using the trapezoid rule we have an approximation which further improves

accuracy of calculation:

)ˆ( Nh )(ˆ)(ˆ2

)(ˆ hzNzNh

zdzNhz

z

(3.11)

),()]ˆ2

exp()(ˆ)(ˆ2

exp)ˆ2

[exp(),( tzALh

hzNzNh

Lh

ThzA

(3.12)


Page | 33

Linear part solution: Step1

)ˆ2

exp( Lh

)](ˆ2

exp[ iLh

and A(z,t) ),( 0 zA

(3.13)

Linear Solution(1) =

),()](ˆ

2exp[ 0

1 zAiL

hFT

(3.14)

Nonlinear part solution

Nonlinear solution )ˆexp( Nh Linear Solution(1) (3.15)

Linear part solution: Step 2

A(z +h ,T) = Nonlinear Solution

),()](ˆ

2exp[ 0

1 zAiL

hFT

(3.15)

Fig 3.2 Symmetrized split step fourier method


Page | 34

3.3 DECISIONS

We had several options before concerning which method to use for solving the

nonlinear Schrödinger equation. In the end we decided to use the symmetrized

split step fourier method. The reasons are given as follows:

The split step fourier method(SSFM) can be used to solve the NLSE

accurately up to the second order dispersion term as relevant to our

dissertation.(See Appendix for Taylor series explanation )

SSFM is faster because the differential terms in the dispersion (linear) part

is calculated in the fourier domain using the relation it

.

The symmetrized SSFM is used because it further improves the accuracy

of the procured results depending on the selected value of the step size h.

The FFT method is used to fourier transform back and forth in the

symmetrized SSFM which reduces computation time because FFT is the

one of the fastest methods available for transformation.

We solved two different forms of the nonlinear Schrödinger equation, the

simplified NLSE and the normalized NLSE, using two different MATLAB

codes. This is done because solution of gaussian pulses and supergaussian

pulses compared to hyperbolic secant pulses require different parameters and

conditions.

CHAPTER 4: IMPLEMENTATION

35 | P a g e

CHAPTER 4

IMPLEMENTATION


36 | P a g e

4.1 NONLINEAR SCHRODINGER EQUATION (NLSE)

SOLUTIONS

4.1.1 Generalized NLSE

)(1

262

2

2

0

2

3

3

3

2

2

2

T

ATAA

TA

iAAiA

T

A

T

Ai

z

AR

(4.1)

4.1.2 SIMPLIFIED NLSE

AAAT

Ai

z

A 2

2

2

2

22 (4.2)

In the previous equations the generalized NLSE is simplified as the higher order

nonlinear effects of self steepening and Raman scattering are removed.

0)(1

2

2

0 T

ATiAA

TAR

(4.3)

Since we have chosen our initial pulse width in order of picoseconds( psT 10 )

this assumption is feasible.


37 | P a g e

We have also considered 3 =0 .For the third order dispersion (TOD) to have

sufficient impact on nonlinear propagation , the optical wavelength must be close

to the zero-dispersion wavelength. In this case 02 which causes TOD to take

effect.

Hence, the criteria we have chosen ensures that 2 is the dominant dispersion

effect thereby justifying removal of the TOD from the NLSE.

4.1.3 NORMALIZED NLSE

02

22

2

2

UUiNUs

iZ

U

(4.4)


38 | P a g e

4.2 GAUSSIAN IMPLEMENTATION

Table 4.1 Gaussian input

Fourier spectrum = TF (U(0, ))

Linear dispersive part (1st half)

Linear solution(1) = TF (U(0, ))

222exp

2

2 hi

Inverse fourier transform:

Linear solution(1) in time domain = 1

TF [ Linear solution(1)]

Nonlinear part

Nonlinear solution in time domain= hi2

domain in time )solution(1Linear exp

Nonlinear part output =Linear solution(1) in time domain Nonlinear solution in time domain

GAUSSIAN PULSE INPUT SIMPLIFIED NLSE METHOD

2

2

2

)1(exp*),0(

to

TiCAoU AAA

T

Ai

z

A 2

2

2

2

22

SSSFM


39 | P a g e

Fourier transform

Nonlinear part output fourier domain=TF [Nonlinear part output]

Linear dispersive part(2nd

half)

Linear solution(2) = Nonlinear part output fourier domain222

exp2

2 hi

Inverse fourier transform

Linear solution(2) in time domain= 1

TF [Linear solution(2)]

SOLUTION FOR NLSE OVER 1 STEPSIZE h = Linear solution(2) in time domain

This process is repeated over the length of the fiber.

Table 4.2 Supergaussian input

The Gaussian implementation method can also be applied for supergaussian pulse.

SUPER GAUSSIAN PULSE INPUT SIMPLIFIED NLSE METHOD

2

2

2

)1(exp*),0(

to

TiCAoU

m

AAAT

Ai

z

A 2

2

2

2

22

SSSFM


40 | P a g e

4.3 SOLITON IMPLEMENTATION

Table 4.3 Hyperbolic Secant input

Fourier spectrum = TF (U(0, ))

Linear dispersive part (1st half)

Linear solution(1) = TF (U(0, ))

222exp

2 his

Inverse fourier transform:

Linear solution(1) in time domain = 1

TF [ Linear solution(1)]

Nonlinear part

Nonlinear solution in time domain= hiN22 domain in time )solution(1Linear exp

Nonlinear part output =Linear solution(1) in time domain Nonlinear solution in time domain

HYPERBOLIC SECANT PULSE

INPUT

NORMALIZED NLSE METHOD

SSSFM

TohNU sec),0(

02

22

2

2

UUiNUs

iZ

U


41 | P a g e

Fourier transform

Nonlinear part output fourier domain=TF [Nonlinear part output]

Linear dispersive part(2nd

half)

Linear solution(2) = Nonlinear part output fourier domain222

exp2 his

Inverse fourier transform

Linear solution(2) in time domain= 1

TF [Linear solution(2)]

SOLUTION FOR NLSE OVER 1 STEPSIZE h = Linear solution(2) in time domain

This process is repeated over the length of the fiber.


42 | P a g e

4.4 FLOW CHART


43 | P a g e


44 | P a g e


45 | P a g e

4.5 PULSE BROADENING RATIO

In this thesis, analysis is done by using the pulse broadening ratio of the evolved pulses. Pulse

broadening ratio is calculated by using the Full Width at Half Maximum.

Pulse broadening ratio=FWHM of propagating pulse / FWHM of First pulse.

Pulse broadening ratio signifies the change of the propagating pulse width compared to the pulse

width at the very beginning of the pulse propagation.

4.5.1 FULL WIDTH HALF MAXIMUM

At the half or middle of the pulse amplitude, the power of the pulse reaches maximum. The

width of the pulse at that point is called full width half maximum.

Fig 4.4: Full Width at Half Maximum


Page | 42

4.4 FLOW CHART

YES

NO

Fig 4.1: Gaussian Flowchart

STOP

splitstep=Z

INPUT FWHM

for splitstep , from h to Z by h

(splitstep=h:h:Z)

fourier.spectrum=fft(A)

f = ifft(fourier.spectrum)

222exp*..

2

2 hispectrumfourierspectrumfourier

fourier.spectrum= fft(f)

loop = 1

INPUT

2

2

2

)1(exp*),0(

To

iCAoA

22exp*..

2

2 hi

spectrumfourierspectrumfourier

f=fft(fourier.spectrum)

START

RATIO

loop = loop +1

FWHM(1:loop )

FWHM

saved at

every

interval

SAVING PULSE AT

EVERY INTERVAL

hfiff2

exp*


43 | P a g e

YES

NO

Fig 4.2: Soliton Flowchart

STOP

splitstep=Z

INPUT FWHM

for splitstep, from h to Z by h

(splitstep=h:h:Z)

fourier.spectrum=fft(U)


222exp*..

2 hisspectrumfourierspectrumfourier


loop = 1

INPUT

To

hNU sec),0(

222exp*..

2 hisspectrumfourierspectrumfourier

f=ifft(fourier.spectrum)

x=0

x = x +1

START

RATIO

loop = loop +1

FWHM(1:loop )

FWHM

saved at

every

interval

SAVING PULSE AT

EVERY INTERVAL

hfiNff22exp*

ln

=1


Page | 44

YES

NO

Fig 4.3: Supergaussian Flowchart

STOP

splitstep=Z

INPUT FWHM

for splitstep , from h to Z by h

(splitstep=h:h:Z)

fourier.spectrum=fft(A)


222exp*..

2

2 hispectrumfourierspectrumfourier


loop = 1

INPUT

2

2

2

)1(exp*),0(

To

iCAoA

m

22exp*..

2

2 hi

spectrumfourierspectrumfourier

f=fft(fourier.spectrum)

START

RATIO

loop = loop +1

FWHM(1:loop )

FWHM

saved at

every

interval

SAVING PULSE AT

EVERY INTERVAL

hfiff2

exp*

CHAPTER 5: ANALYSIS OF RESULTS

46 | P a g e

CHAPTER 5

ANALYSIS OF RESULTS


47 | P a g e

ANALYSIS PLAN:

Table 5.1: Test Plan

NUMBER INPUT PULSE

PROPERTIES

1 Gaussian Pulse Pulse broadening ratio for variable group

velocity dispersion parameters 2

2 Gaussian Pulse Pulse broadening ratio for various nonlinear

parameter

3 Gaussian Pulse

Pulse broadening ratio for various Input

Power, I

4 Fundamental Soliton,

Hyperbolic secant pulse

Pulse broadening ratio

5 Higher Order Soliton,

Hyperbolic secant pulse

Pulse broadening ratios for various power of

the soliton order N

6 Gaussian Pulse

Pulse broadening ratio for Chirp, C = 0

7 Gaussian Pulse

Pulse broadening ratio for Chirp ,C = 1

8 Gaussian Pulse

Pulse broadening ratio for Chirp, C = 2

9 Gaussian Pulse

Pulse broadening ratio for Chirp , C = -2

10 Super Gaussian Pulse Pulse Broadening Ratio for different values of

chirp for m=4

11 Super Gaussian Pulse Pulse Broadening Ratio for different values of

chirp for m=100


48 | P a g e

GAUSSIAN PULSE ANALYSIS

Input curve for Gaussian pulse

Fig5.1: Input Gaussian Pulse


49 | P a g e

Pulse broadening ratio for different dispersion parameter, 2 is given below:

Fig5.2: Pulse Broadening Ratio for different values of GVD

5.2.1 Analysis of Beta variation:

In this figure, the pulse broadening ratios of gaussian pulses are observed using variable group

velocity dispersion parameters 2 while keeping constant the power and nonlinear parameter .

The values chosen for this study are 2 = 271020 , 271040 , 271060 2710100 ,

2710500 .

The purpose here is to simply observe what the effect of increasing or decreasing dispersion

parameter will have on pulse broadening and how much. It is seen, that over a length of fiber the

increasing 2 leads to a general increase in the pulse broadening ratio.


50 | P a g e

The physical meaning of this is that the high frequencies (blue shifted) of given optical pulse

travels faster than the low frequencies(red shifted) which causes GVD induced negative chirp in

the anomalous dispersion region . This essentially causes spreading of the pulse. Now increasing

the GVD parameter 2 results the high frequencies to travel faster and the low frequencies to

travel slower. So for a given length of propagation each frequency component of an optical pulse

arriving with larger and larger delay resulting in greater pulse spreading with each increasing

value of2 .

However, as we continue increasing the GVD parameter we notice a property that the difference

of Pulse broadening ratio curves between higher values of 2 , for example 2710500 and

2710100 is lesser than that between 271060 and 271040 .


51 | P a g e

Pulse broadening ratio for various nonlinear parameter is given below:

Fig 5.3: Pulse Broadening Ratio for different values of Nonlinearity

5.2.2 Analysis of Nonlinear parameter variation:

In this figure, we study the importance of magnitude of nonlinear parameter on nonlinear

optical fiber.


52 | P a g e

While keeping the input power and the GVD parameter constant, we generate curves for various

values of , both greater and smaller than the ideal value used for proper GVD and SPM

balance.

The values chosen for this study are = 0.003, 0.006, 0.009, 0.011 and 0.001 /W/m.

The purpose here is to observe what the effect of increasing or decreasing nonlinear parameter

will have on pulse broadening and how much. It is seen, that over a length of fiber the

increasing leads to an initial decrease in the pulse broadening ratio.

For =0.003 , the SPM induced positive chirp gradually cancels out the GVD induced negative

chirp obtaining steady width pulse propagation.

For =0.006, the pulse appears to initially narrow more than the previous case because of

increased nonlinearity hence increased SPM effect. In this case also, the pulse width once

appears to gradually attain towards a constant pulse width. This happens because although

initially SPM effect becomes dominant , as propagation distance increases the GVD effect

increases eventually cancelling out the SPM induced positive chirp.

For = 0.009 and 0.011, the pulse broadening ratio initially decreases to a minimum value. The

minimum value for =0.011 is smaller than for =0.009, which means that the pulse narrows

more for the former than the latter expected. However, as GVD effect becomes more prominent,

in these two cases the pulse does not acquire a constant pulse width further down the propagating

line.

For =0.001, it is obvious that the SPM effect is not large enough counter the GVD effect.

From these observations, we can deduce that for a given 2 , we can increase the value to a

certain point until which we can obtain equal GVD and SPM effect cancellation. Beyond this

value becomes dominant and pulse shape is lost.


53 | P a g e

Figure of pulse broadening ratio for Gaussian pulse with input power =0.00005w, 0.0001w,

0.00064w and 0.001w.

Fig5.4: Pulse Broadening Ratio for different values of Power

5.2.3 Analysis for Power variation:

In this figure, we study the importance of magnitude of power on nonlinear optical fiber.

While keeping the nonlinear parameter and the GVD parameter constant, we generate curves for

various values of input power, both greater and smaller than the ideal value used for proper GVD

and SPM balance. It is observed from this plot, that the pulse broadening ratio for curves with

smaller input power is more than that with larger input power.


54 | P a g e

This property can be explained by the following equation:

o

NP

L

1 ; where, oP is the input power and NL is the nonlinear length.

This equation shows that the nonlinear length is inversely proportional to the input power.

As a result NL decreases for higher values of oP . For oP =0.001W it is observed that the pulse

broadening ratio decreases meaning narrowing of pulses. This is surprising because although the

nonlinear parameter is constant; narrowing of pulses continues to occur.

The reason is that the same amount of nonlinear effect occurs but it manifests itself over NL

which is now shorter and hence narrowing occurs faster than before.

Reducing oP results in an opposite effect. The stays constant but NL is now larger and hence

narrowing occurs over a larger distance and hence slower. So GVD acts quicker and becomes

dominant and so spreading of pulse occurs.


55 | P a g e

5.3 SOLITON PULSE ANALYSIS

Input Soliton pulse:

Fig5.5: Input Hyperbolic secant Pulse

Pulse propagation for fundamental soliton:

Fig5.6: Soliton Pulse Propagation


56 | P a g e

Pulse broadening Ratio for soliton propagation:

Fig5.7: Pulse Broadening Ratio for Soliton Pulse Propagation

Soliton pulse Analysis:

Hyperbolic secant pulse propagation without initial chirp is simulated by analytically solving the

normalized Schrodinger equation using the split step fourier method. From Figure 5.6 we

observe that the pulse propagates at seemingly constant width. From the curve in Figure 5.7 the

pulse broadening ratio is found to be a steady, horizontal line confirming that the pulse travels at

a constant width which is equal to the input width of the pulse. This propagation of a constant

width hyperbolic secant pulse means that we have obtained soliton propagation in nonlinear

optical fiber.


57 | P a g e

Soliton propagation is possible under certain conditions. In general, the dispersion length,DL

must be approximately equal to the nonlinear length, NL . This would mean that the group

velocity dispersion would take effect over the same length as the nonlinear effects.

Under such circumstances, the GVD effect matches the SPM effects entirely and cancels each

other out to obtain steady pulse width throughout the length of the fiber. We assume that no

attenuation is present for simplification of solution.

The GVD and the SPM manifest themselves through their induced chirp effects. For GVD, the

high frequency components of the pulse travel at higher velocity than the low frequencies which

induces negative chirp causing dispersion. SPM on the other hand, induces positive chirp during

propagation causing the pulse to narrow as it evolves.

When ND LL , the negative induced chirp resulting from the GVD and the positive induced

chirp caused by the SPM become equal in magnitude and opposite in direction.

The net outcome is that the negative induced chirp of GVD cancels out the positive induced

chirp of SPM equally producing fundamental soliton (N=1).


58 | P a g e

5.4 HIGHER ORDER SOLITON ANALYSIS

Pulse propagation for higher order soliton for N=2.



59 | P a g e






60 | P a g e

Pulse Broadening Ratio for Higher Order Soliton Pulse Propagation with N=2, 3, 4 and 5:

Fig5.11: Pulse Broadening Ratio for Higher Order Soliton with different N

Higher order soliton Analysis:

Here, the power of the soliton order N is increased gradually to obtain pulse broadening ratios of

higher order solitons. These ratios are plotted on the same axis for comparison of pulse

propagation of each of the higher order solitons for a given length of fiber. The values chosen for

N are 2,3,4,5 and 10.

2

2

000

2

TPLPN D


61 | P a g e

The previous equation shows N2 which is the nonlinear factor used for the solution of the time

domain nonlinear part of the normalized nonlinear Schrodinger equation. The input to the NLSE

is:

U(0, )=Nsech0T

So increasing the value of N not only affects the magnitude of the input but also influences the

nonlinear solution of the NLSE resulting in the varied behavior of each higher order soliton.

The second order soliton (N=2, green line) produces a periodic outcome. It is observed that after

a certain period of traversed length the initial pulse shape is re-acquired. This pattern continues

to occur at regular intervals. The length of propagating distance after which the initial pulse

shape is re-obtained is called the soliton period and is given by the following equation:

2

2

00

22

TLz D

The third order soliton (N=3, Blue line) also produces a periodic pattern in the pulse broadening

ratio curve. At regular intervals, the width of the pulse oscillates; however it does not seem to

re-acquire the original pulse shape but slowly heads towards acquiring a different pulse width.

The fourth (red line) and fifth order (black line) oscillates for shorter and shorter period of

length. At the end of the chosen propagating distance, the pulse broadening ratio seems to appear

less oscillatory and heading towards achieving constancy at a much lower value.


62 | P a g e

However, this event is characterized by splitting of pulses along the length of propagation.

(Waterfall figures of higher order pulse evolution attached with appendix).

The reason for the behavior of higher order soliton is that the pulse splits into several small

pulses. The net effect is the eventual destruction of the intended information due to

fragmentation of the pulses.


63 | P a g e

5.5 CHIRPING EFFECT ANALYSIS

Pulse broadening ratio of Gaussian pulse with chirp, C=2, 0, -1, -2.

Fig5.12: Pulse Broadening Ratio of Gaussian Pulse with different chirp


64 | P a g e

Figure of Pulse Propagation of Gaussian pulse for chirp 0


5.5.1 Analysis for Un-chirped Gaussian pulse

Here, both GVD and SPM act simultaneously on the propagating gaussian pulse with no initial

chirp. As can be seen from the evolution graph, the pulse shrinks initially for a very small period

of propagating length. After that the broadening ratio reaches a constant value and a stable pulse

is seemed to propagate as can be seen from pulse broadening ratio curve.

GVD acting individually results in the pulse to spread gradually before it loses shape. SPM

acting individually results in the narrowing of pulses and losing its intended shape.


65 | P a g e

The combined effect of GVD and SPM leads to the eventual generation of constant pulse

propagation emulating a hyperbolic secant pulse.

Initially, the positive chirp induced by SPM seems to dominate the negative chirp caused by the

GVD which accounts for the narrowing of the propagating pulse in the early steps. But after a

certain period, GVD effect becomes more prominent and it catches up to the SPM effect. At one

point, GVD induced negative chirp balances the SPM induced positive chirp, and they cancel

each other out and propagates at a narrower pulse width emulating hyperbolic secant pulse

propagation.


66 | P a g e

Figure of Pulse Propagation of Gaussian pulse for chirp -1


5.5.2 Analysis for Gaussian pulse of Chirp = -1

Here, both GVD and SPM act simultaneously on the gaussian pulse with initial negative chirp.

The evolution pattern shows that pulse broadens at first for a small period of length. But

gradually the rate at which it broadens slowly declines and the pulse broadening ratio seems to

reach a constant value. This means that the pulse moves at a slightly larger but constant width as

it propagates along the length of the fiber. Although the width of the pulse seems constant, it

does not completely resemble a hyperbolic secant pulse evolution. We compare the pulse

evolution of the gaussian pulse with no initial chirp in Figure 5.13 and the negative chirped


67 | P a g e

gaussian pulse evolution in Figure 5.14 to see the difference in shape and width of each of these

evolutions.

As we previously established GVD and SPM effects cancel each other out when the GVD

induced negative chirp equals the SPM induced positive chirp. But in this case the initial chirp

affects the way both GVD and SPM behave. The chirp parameter of value -1 adds to the negative

chirp of the GVD and deducts from the positive chirp of SPM causing the net value of chirp to

be negative. This means that GVD is dominant during the early stages of propagation causing

broadening of the pulse.

But as the propagation distance increases the effect of the initial chirp decreases while the

induced chirp effect of both GVD and SPM regains control. The difference between positive and

negative induced is lessened and just like in the case of gaussian pulse propagation without

initial chirp the GVD and SPM effects eventually cancels out each other to propagate at constant

width.


68 | P a g e

Figure of Pulse Propagation of Gaussian pulse for chirp -2


5.5.3 Analysis for Gaussian pulse of Chirp = -2

Here, both GVD and SPM act simultaneously on the gaussian pulse with initial negative chirp.

From Figure 5.15 we observe that the pulse width broadens as it propagates. From the pulse

broadening ratio curve of the Figure 5.12 we see that the pulse broadens from the very start and

continues to broaden. It is also obvious that the rate at which the pulse broadens is higher than in

previous cases.

As before we concern ourselves with anomalous dispersion region where 2 < 0.


69 | P a g e

Therefore for C= -2 we have 02 C which as explained previously means that the direction of

dispersion induced chirp is in the same direction to that of the initial chirp value.

The negative input chirp C adds to the negative induced chirp of the GVD. At the same time the

positive chirp caused by SPM is reduced due to deduction from the negative input chirp. Hence,

as the increasing GVD effect causes dispersion of the pulse at an enlarged rate, the curtailing of

the SPM effect results in the reduction of the rate at which the pulse narrows. The net effect is

that the rate of dispersion is much higher than the rate of pulse narrowing which explains why

the pulse broadening ratio increases at a much higher rate than previous cases.

Also, unlike in the case of chirp 0 and chirp -1, the impact of the initial chirp parameter is greater

if the magnitude of the chirp is more than a certain value. In such a case, the GVD and SPM do

not cancel each other further down the length of the propagating medium; instead the pulse

continues to broaden indefinitely. This value of the initial chirp where GVD effect grows and

eventually dominates pulse propagation is the CRITICAL chirp. Beyond such a value, the

transformation of the gaussian pulse into a constant width pulse is impossible.


70 | P a g e

Figure of Pulse Propagation of Gaussian pulse for chirp 2


5.5.4 Analysis for Gaussian pulse of Chirp = 2

Here, both GVD and SPM act simultaneously on the gaussian pulse with initial positive chirp.

The pulse evolution graph in Figure 5.16 describes a gaussian pulse in nonlinear media with an

initial chirp of positive 2. Here, the input pulse narrows substantially in the early goings of pulse

evolution, but only for a certain length of the fiber. Following its brief narrowing tendency

during which it seems to gain amplitude height, the pulse seems to spread as it propagates. The

pulse broadening ratio curve in Figure 5.12 confirms the aforementioned tendency and the fact

that the ratio rises sharply after its initial fall.


71 | P a g e

In this case since, C>0, so that 02 C . This means that the dispersion induced chirp is in

opposite direction to that of the initial chirp parameter C.

The positive input chirp deducts from the negative induced chirp and results in a decreased

dispersive effect. Simultaneously, the positive input chirp adds to the SPM induced positive

chirp. The net outcome is that the dispersive effect due GVD declines and the SPM effect rises,

explaining the initial narrowing of the pulse width.

However, as the distance of propagation increases the GVD effect gains on the SPM effect and

eventually becomes dominant. Eventually, the pulse broadening ratio begins increasing linearly.

The outcome is once again attributed to the initial chirp being greater than the critical chirp

permissible for gaussian pulses.


72 | P a g e

5.6 SUPER GAUSSIAN PULSE ANALYSIS

Input pulse of super Gaussian for m=1, 8, 40 and 100:

Fig5.17: Input pulse for different values of m.


73 | P a g e


Fig5.19 Supergaussian pulse propagation for m=10


74 | P a g e




75 | P a g e

Pulse Broadening Ratio:

Fig5.22: Pulse Broadening Ratio for different values of m for chirp 0

5.6.1 Analysis for Supergaussian pulse for chirp =0:

Supergaussian pulse broadening ratios for various powers of supergaussian pulses are studied. In

Figure: 5.22 we have considered zero initial chirp and zero nonlinearity to learn of the effect of

dispersion for higher power supergaussian pulses. From here we obtain pulse broadening ratio

curves for m = 4, 10, 25, 50 and 100. The behavior of the pulses is easily viewed. As we

continue increasing the power m of the supergaussian, the slopes of the straight lines of each of

the curves increase. This suggests that the steeper the edge of the supergaussian pulses higher are

the effects of GVD induced negative chirp. Hence, supergaussian pulses of higher power will

experience greater dispersive effects than that of lower power supergaussians.


76 | P a g e

Fig5.23: Pulse Broadening Ratio for different values of chirp for m=100

5.6.2 Analysis for Supergaussian pulse for chirp =100:

In Figure: 5.23 we consider supergaussian pulses of power m= 100 to study the effects under

various input chirp parameters. The nonlinearity is considered zero here.

In this case, chirp =0, 1, 2 and -2 are studied. The pulse broadening ratio curves reveal that as

the magnitude of chirp is increased from 0 to 1, the effect of dispersion seems to increase. But as

chirp is increased from 2 and then decreased to -2, the effect of dispersion seems to decrease and

appears to reach a constant level. This is significant because here keeping the magnitude of the

chirp relatively large for supergaussian pulses of high power can reduce dispersive effects.

CHAPTER 6: CONCLUSION

Page | 77

CHAPTER 6

CONCLUSION

In this dissertation we explored the combined effects of the GVD and SPM on various types of

pulses including hyperbolic secant pulses, gaussian pulses and supergaussian pulses. At first ,

we propagate a gaussian pulse and observe results for variable nonlinearity, variable group

velocity dispersion and variable input power in three separate studies. We find that for low

nonlinear parameter values the pulse regains initial shape for a given input power and 2 .But

for higher values pulse collapses. By altering 2 for same input power and nonlinear parameter,

we observe that the pulse broadening ratio increases for higher values of 2 . Increasing input

power while keeping 2 and nonlinear parameter constant , we find that nonlinear effects

increases. Decreasing input power results in decrease in nonlinear effect , so that GVD becomes

dominant and pulse disperses.

Hyperbolic pulses are propagated as a constant width pulse called soliton. The perfect

disharmonious interaction of the GVD and SPM induced chirps result in diminishing of both

dispersive and nonlinear narrowing effects and hence soliton is obtained. Gaussian pulses are

also propagated with or without pre-induced(initial) chirp to study the pattern of propagation. It

is found that in the case of chirp 0 and chirp -1 , the gaussian pulse acquires a hyperbolic secant

pulse shape and travels as a pseudo-soliton. However, higher values of initial chirp leads to

indefinite dispersion and pulse shape is not retained ; a fact that can be attributed to the critical

chirp, a chirp value beyond which no constant width pulse propagation is possible. For the super-

gaussian pulse propagation we first considered an unchirped input with zero nonlinearity

parameter to understand the effects of the power of the super-gaussian pulse m on the pulse

width and found that pulse broadening ratio curve becomes steeper for higher powered super-

CHAPTER 6: CONCLUSION

Page | 78

gaussians. We then applied initial chirp on the super-gaussian pulses and found that for values of

chirp 2 and -2 or higher , the high power super-gaussian pulse broadening steadies signifying a

decrease in dispersive effects. We also generated pulse broadening ratio curves and evolution

patterns for higher order solitons. Here , we find that as we increase the soliton order, for N=2

the pulse width periodically varies and regains the original pulse after soliton period oz .But

increasing the soliton order further results in initial periodic behaviour of pulse width before

settling at a much lower value indicating that pulse splitting has occurred.

BIBLOGRAPHY

Page | 79

Bibliography

1. Dutta, A. K. WDM Technologies - Active Optical Components, 2002

2. G. P. Agrawal, Fiber-Optic Communication Systems, Second Ed., John Wiley &

Sons, Inc. New York, 1997

3. G. P. Agrawal, Nonlinear Fiber Optics, Second Ed., Academic Press, San Diego,

1995

4. Agrawal, G. P. Applications of Nonlinear Fiber Optics, 2001.

5.Becker, P. C. Erbium-Dope Fiber Amplifiers - Fundamentals and Technology, 1997.

6. M.J. Ablowitz , H. Segur, Solitons and the inverse Scattering Transform, Siam,

Philadelphia,1998.

7. Oleg V. Sinkin, Ronald Holzlöhner, John Zweck,„„Optimization of the Split-Step Fourier

Method in Modeling Optical-Fiber Communications Systems‟‟, in Journal of lightwave

technology , January, 2003

8. Decusatis,, C. Handbook of Fiber Optic Data Communication ,2nd ed ,2002.

9. Vinay M K, „„Dispersion Managed Solitins‟‟. Department of Electrical Engineering

IIT Madras,May 2005

10. Kaminow, I. P. Optical Fiber Telecommunications III-A, 1997.

11. G. M. Gharib, „„soliton Solutions for the Unstable Nonlinear Schr¨odinger Equation which

Describe Surface of Constant Negative Curvature‟‟, in Int. Journal of Math. Analysis, Vol. 2,

2008,

12. Eran Bouchbinder, „„The Nonlinear Schr¨odinger Equation‟‟,2003.

13. Mihajlo Stefanović, Petar Spalevic, Dragoljub Martinovic, Mile Petrovic,„„Comparison of

Chirped Interference Influence on Propagation Gaussian and Super Gaussian Pulse along the

Optical Fiber‟‟, in journal of Optical Communications,2006

14. P. Lazaridis, G. Debarge, and P. Gallion, „„Optimum conditions for soliton launching from

chirped sech 2 pulses”, in optics letters, August 15, 1995

15. C. C. Mak, K. W. Chow, K. Nakkeeran, „„Soliton Pulse Propagation in Averaged Dispersion-

managed Optical Fiber System”, in Journal of the Physical Society of Japan, May, 2005,

BIBLOGRAPHY

Page | 80

16. E. Iannone, et al., Nonlinear Optical Communication Networks, John Wiley &

Sons, Inc., 1998.

17. A. Andalib, A. Rostami, N. Granpayeh, ‘‘Analytical investigation and evaluation of pulse

broadening factor propagating through nonlinear optical fibers‟‟, in Progress In

Electromagnetics Research, PIER, 2008

18. .Crisp, J. Introduction to Fiber Optics (2nd ed.), (2001).

19. “Soliton pulse propagation in optical fiber‟‟, class notes for WDM and optical Networks

course, MIT Lincoln Laboratory, December,2001.

20. Decusatis, C. Fiber Optic Data Communication - Technological Trends and Advances,

(200121. J.C. Malzahn Kampe, MATLAB Programming,2000.

GLOSSARY

Page | 81

MATLAB CODES:

GAUSSIAN CODE

clc; clear all; close all; clf;

loop=1;

i=sqrt(-1);

pi=3.1415926535;

Po=0.00064; %input pwr in watts

alpha=0; % Fiber loss value in dB/km

Ao=sqrt(Po); %Amplitude of pulse

gamma=0.003; %nonlinear parameter in /W/m

To=125e-12; %initial pulse width in seconds

beta2=-20e-27; %2nd order disp. (s2/m)

Lengthdisp=(To^2)/(abs(beta2)); %dispersion length in

meter

chirp=0; %Input chirp parameter

%----------------------------------------------------------

tau =- 4096e-12:1e-12: 4095e-12;% dt=t/to

delt=1e-12;

rel_error=1e-5;

h=1000;% step size

for Lengthprop=0.1:0.1:1.5

GLOSSARY

Page | 82

z=Lengthprop*Lengthdisp;

A=Ao*exp(-((1+i*(chirp))/2)*(tau/To).^2);

figure(1)

plot(abs(A));

title('Input Pulse');

xlabel('Time');

ylabel('Amplitude');

grid on;

hold on;

amp=max(size(A));

%---------------------------------------------------------------------

-----

input.fwhm=find(abs(A)>abs(max(A)/2));

input.fwhm=length(input.fwhm);

domega=1/(amp)/delt*2*pi;

omega=(-1*amp/2:1:amp/2-1)*domega;

A=fftshift(A);

omega=fftshift(omega);

fourier.spectrum=fft(fftshift(A)); %Pulse spectrum

for splitstep=h:h:z

Linsoln1=exp(-alpha*(h/2)+i*beta2/2*omega.^2*(h/2)) ;

fourier.spectrum=fourier.spectrum.*Linsoln1;

f=ifft(fourier.spectrum);

Nonlinsoln=exp(i*gamma*((abs(f)).^2)*(h));

f=f.*Nonlinsoln;

fourier.spectrum=fft(f);

GLOSSARY

Page | 83



end


pulse.output(loop,:)=abs(f);%saving output pulse at all intervals

output.fwhm=find(abs(f)>abs(max(f)/2));

output.fwhm=length(output.fwhm);

ratio=output.fwhm/input.fwhm; %PBR at every value

pbratio(loop)=ratio;%saving PBR at every step size

loop=loop+1;

end

figure(2);

waterfall(pulse.output);

title('Pulse Evolution');

xlabel('Time');

ylabel('Distance');

zlabel('Amplitude');

figure(3)

GLOSSARY

Page | 84

plot(pbratio(1:1:loop-1),'b');

xlabel('Total number of steps:Each step is 1 SSFM cycle');

ylabel('Pulse broadening ratio');

grid on;

hold on;

SOLITON CODE:


loop=1;

i=sqrt(-1);

signbeta=-1;% sign of beta2

alpha=0; % Fiber loss

nonlinpar=0.003; %fiber non linearity in /W/m

N=1; %soliton order

To=125e-12; %initial pulse width in second

pi=3.1415926535;


Ao=sqrt(Po); %Amplitude

Lengthdisp=(N^2)/(nonlinpar*Po);%dispersion length for

corresponding soliton order

beta2=-(To)^2/Lengthdisp; %2nd order disp. (s2/m)

chirp=0;

Nfactor=N.^2;

GLOSSARY

Page | 85

tau =- 4096e-12:1e-12: 4095e-12;% dt=t/to

delt=1e-12/To;

hnorm=1000;% step size

%-------------------------------------------------------------------------

%Defining interval for distance using SECH pulse

%-------------------------------------------------------------------------



%--------------------------------------------------------------------------

%FUNDAMENTAL SOLITON

U=N*sech(tau/To);

%--------------------------------------------------------------------------

%--------------------------------------------------------------------------

h=hnorm/Lengthdisp;

znorm=z/Lengthdisp;

amp=max(size(U));

%--------------------------------------------------------------------------

input.fwhm=find(abs(U)>abs(max(U)/2));


fourier.spectrum=fft(fftshift(U)); %fourier domain pulse spectrum

domega=(1/amp)/delt*2*pi;



x=0;

GLOSSARY

Page | 86

%----------------------------------------------

%Split Step fourier method

%----------------------------------------------

for splitstep=h:h:znorm

Linsoln1=exp(-alpha*(h/2)+i*(signbeta)/2*omega.^2*(h/2));



Nonlinsoln=exp(i*(Nfactor)*((abs(f)).^2)*(h));

f=f.*Nonlinsoln;

f=fftshift(f);


Linsoln2=exp(-alpha*(h/2)+i*(signbeta)/2*omega.^2*(h/2)) ;


x=x+1;

end


f=fftshift(f);





GLOSSARY

Page | 87


loop=loop+1;

end

figure(1)

plot(abs(U));

xlabel('Time');ylabel('Amplitude');

grid on;

hold on;

figure(2);


xlabel('Time');

ylabel('Distance');


grid on;

hold on;

figure(3)

plot(pbratio(1:1:loop-1),'k');



grid on;

hold on;

GLOSSARY

Page | 88

SUPERGAUSSIAN CODE:


loop=1;

i=sqrt(-1);

pi=3.1415926535;


alpha=0; % Fiber loss value in dB/km

Ao=sqrt(Po); %Amplitude of pulse

gamma=0.003; %nonlinear parameter in /W/m

To=125e-12; %initial pulse width in seconds

beta2=-20e-27; %2nd order disp. (s2/m)

Lengthdisp=(To^2)/(abs(beta2)); %dispersion length in meter

chirp=0; %Input chirp parameter

m=4; % Power of the super gaussian pulse

%----------------------------------------------------------

tau =- 4096e-12:1e-12: 4095e-12;% dt=t/to

delt=1e-12;

rel_error=1e-5;

GLOSSARY

Page | 89

h=1000;% step size



A=Ao*exp(-((1+i*(chirp))/2)*(tau/To).^2*m);

figure(1)

plot(abs(A));

title('Input Pulse');

xlabel('Time');

ylabel('Amplitude');

grid on;

hold on;

amp=max(size(A));

%--------------------------------------------------------------------------

input.fwhm=find(abs(A)>abs(max(A)/2));


domega=1/(amp)/delt*2*pi;


A=fftshift(A);


fourier.spectrum=fft(fftshift(A)); %Pulse spectrum

GLOSSARY

Page | 90

for splitstep=h:h:z




Nonlinsoln=exp(i*gamma*((abs(f)).^2)*(h));

f=f.*Nonlinsoln;




end







loop=loop+1;

end

GLOSSARY

Page | 91

figure(2);


title('Pulse Evolution');

xlabel('Time');

ylabel('Distance');


figure(3)

plot(pbratio(1:1:loop-1),'b');



grid on;

hold on;

GLOSSARY

Page | 92

LIST OF SYMBOLS

A(z,T) Slowly varying pulse envelope

oA sqrt ( oP )

effA Effective core area of fiber

2A Energy of the input pulse

c Speed of light

C Initial chirp parameter

D Dispersion parameter ; D =22

2

c

h stepsize of each SSFM cycle

DL Dispersion length,

2

2

0

TLD

NL Nonlinear length, o

NP

L

1

N soliton order

Nfactor Nfactor=

2

2

000

2

TPLPN D

2n Nonlinear index co-efficient

oP Input power (W)

s sgn ( 2 )

GLOSSARY

Page | 93

T Time parameter for NLSE

0T Initial pulse width

U(z, ) Normalized slowly varying envelope

z propagation distance for simplified NLSE

Z propagation distance for normalized NLSE, Z=DL

z

0z soliton period,

2

2

00

22

TLz D

Fiber attenuation (loss) in dB

2 Second order group velocity dispersion parameter

Nonlinear parameter, effcA

n 02

(T) Induced chirp

Time parameter for normalized NLSE, = 0T

T

0 Center frequency

Numerical Analysis of Nonlinear Pulse Propagation in Optical Fiber

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