ministry of higher education and scientific research ... · modulation , fm-harmonic mode-locking...

Republic of Iraq

Ministry of Higher Education and Scientific Research

University of Technology

Laser and Optoelectronics Engineering Department

Study of the

characterization

design of mode-locked

fiber laser

A Thesis Submitted to

The Laser and Optoelectronics Engineering Department,

University of Technology in Partial Fulfillment of the

Requirements for the Degree of Master of Science in

Laser Engineering

By

Salam Sami Mohammed-Salih B. Sc. Electrical Engineering

1982

Supervised by

Dr. Walled Y. Hussen

Rabia Thani 1429 A. H. April 2008 A. D.

عراقجمھورية ال مي لوزارة التعليم العالي والبحث الع

الجامعة التكنولوجية قسم ھندسة الليزر والبصريات االلكترونية

نماطألدراسة خواص تصميم ا

ليزر االلياف البصريةل المقفلة

رسالة مقدمة الى قسم ھندسة الليزر والبصريات االلكترونية الجامعة التكنولوجية

الليزر نيل درجة الماجستير علوم في ھندسة من متطلبات كجزء

المھندس تقدم بھا

صالح سالم سامي محمد بإشراف

الدكتور وليد ياسين حسين

ھ1429 ربيع الثاني م 2008 نبسان

II

Abstract

In this work, a study of Mode-Locked fiber laser is done. The

study is focusing on Active Mode-Locking by investigating frequency

modulation , FM-Harmonic Mode-Locking for Ytterbium Doped Fiber

Laser as the field of this research. The model studied , uses ytterbium-

doped, single mode fiber, operating with 1055 nm wavelength with 976

nm optical pump and FM-Harmonically Mode-Locked by MZI optical

modulator. A dispersion compensation technique (grating pair) is used.

Mode-Locking fiber laser master equation is introduced, which is

essentially Generalized Nonlinear Schrödinger equation GNLSE. By

using GNLE (or modified Ginzburg-Landu equation), and applying the

moment method, a set of ordinary differential equations is introduced.

These equations describe pulse parameters evolution in dispersive-

nonlinear medium .A numerical solution for these equations using fourth-

fifth order, Runge-Kutta method is performed through MatLab 7.0

program. The effect of both normal and anomalous dispersion regimes on

output pulses is investigated. Also, modulation frequency effect on pulse

parameters is investigated by driving the modulator into different values

of frequencies. This study shows the stability of working in anomalous

dispersion regime and good pulse compression effect due to the

combination effect of both Group Velocity Dispersion, GVD and

nonlinearity .Also it shows the great effect of modulation frequency on

all pulse parameters and stability of the system , and an increase in the

repetition rate .

III

Table of Contents

I AcknowledgmentII Abstract III Table of ContentsVII List of abbreviations IX List of Symbols

List of Figures XII

Chapter One

Ultra-Short Pulses Generation

1.1 Introduction 1 1.2 Ultra-Short Pulses Generation Techniques 3

1.2.1 Q -Switching 3 1.2.2 Mode-Locking 5 1.3 Historical Background 7 1.4 Aim of the Thesis 8 1.5 Thesis Layout 9

Chapter Two

Mode-Locking Fiber Lasers

2.1 Introduction 10 2.2 Fiber Laser 14 2.2.1 Factors affect on Fiber Laser Characteristics 16 2.2.2 Rare-Earth Doped Fibers Significant Properties 16

2.2.3 Disadvantages of Mode-locked Fiber Laser Systems 17

2.2.4 Ytterbium Doped Fibers 18

2.3 Losses, Dispersion and Nonlinearity in Doped Fiber 20

2.3.1 Fiber Losses 20

IV

2.3.1.1 Material Absorption 21

2.3.1.2 Rayleigh Scattering 23

2.3.1.3 Bending Losses 24

2.3.2 Dispersion 25

2.3.2.1 Chromatic Dispersion 26

2.3.2.1.1 Material Dispersion 27

2.3.2.1.2 Normal and Anomalous Dispersion 30

2.3.2.1.3 Waveguide Dispersion 32

2.3.2.1.4 Polarization Mode Dispersion (PMD) 33

2.3.3 Non-Linear Effects in Optical Fibers 33

2.4 Mode-Locked Lasers 35

2.4.1 Physics of Mode Locking 36

2.4.2 Parameters Limiting Pulse Duration 40

2.4.3 Time-Bandwidth Product 40

2.4.3 Mode Locking Techniques 41

2.4.3.1 Active Mode Locking 41

2.4.3.1.1 AM Mode Locking 42

2.4.3.1.2 FM Mode-Locking 42

2.4.3.2 Passive Mode Locking 44

2.4.3.3 Hybrid Mode Locking 44

Chapter Three

The Model of Mode-Locked Fiber Laser

3.1 Pulse Propagation in Optical Fibers 45

3.1.1 Maxwell Equations 45

3.2 Mode-locking Fiber laser Master Equation 49

V

3.2.1 Identifying Mode locking Master Equations Terms 49

3.3 FM Mode-Locking Significant Properties 51

3.3.1 FM Mode-Locker Effect 52

3.4 Moment Method 52

3.5 Pulse Parameters Evolution Equations 53

3.6 Block diagram Model 55

3.7 Mat-Lab Program Design

56

Chapter Four

Numerical Solution and Results

4.1 Normal Dispersion Regime 60

4.2 Anomalous Dispersion Regime 67

4.3 Effect of Changing Modulation Frequency on Pulse Parameters

73

4.3.1 Variable Modulation Frequency Effect in Normal

Dispersion

73

4.3.2 Pulse Parameters Modulation Frequency dependency 82

4.3.3 Variable Modulation Frequency effect in Anomalous

Dispersion

84

4.3.4 Pulse Parameters versus Modulation Frequency for

both Dispersion Regime

94

4.3.4.1 Pulse Energy Modulation Frequency Dependency 94

4.3.4.2 Pulse Temporal Shift Modulation Frequency

Dependency

96

4.3.4.3 Pulse Frequency Shift Modulation Frequency

Dependency

96

VI

4.3.4.4 Pulse Chirp Modulation Frequency Dependency 97

4.3.4.5 Pulse Width Modulation Frequency Dependency 97

Chapter Five

Conclusions and Suggestion for Future Work

5.1 Conclusion 99

5.2 Suggestion for Future Work 101

References 102

IX

List of Symbols Symbol Meaning Symbol Meaning

A t, z Slowly Varying Envelope of the Electric Field

α m Attenuation Constant

a m Core Radius α m Average Losses a Pulse Amplitude Deg. Incident Critical

Angle B Strength of jth

Resonance dB/km Attenuation in dB

Magnetic Flux Density Vector

αR dB/km Rayleigh Scattering Losses

b m Optical Fiber Outer Radius β

Mode-Propagation Constant

c m/s Light Velocity β s/m First Order Dispersion

βfs

dB/km-μm Rayleigh Scattering Constant

/m

Second Order Dispersion

D ps/km.nm Dispersion βParameter

fs m

Averaged Second / Order Dispersion

ux β fs

Electric FlDensity Vector /m

Third Order Dispersion

L m βCavity Length fs

hird /m

Averaged TOrder Dispersion

E Field Amplitude γ m W

Average Non-Linearity

E pJ y ∆dex

Peak Pulse Energ Relative Core-cladding InDifference

Electric Field Vector

∆ pJ en nd

E Variation BetweMaximum aMinimum Energy Value

F GHz Modulation Frequency

∆FM Modulation Depth

F r

Frequency

∆ GHzFrequency Spacing

GHz Repetition oModulation

Axial Modes

X

Sy Symbmbol Meaning ol Meaning

g m ain ∆ ns Roundtrip Duration =Saturation G(1/∆ )

g m Average small- ∆signal gain

t ,

Shortest pulse ns duration

Magnetic Field ∆ rad/fs n M ium idth

aximum Vector

Gai edSpectral Full Wat Half-M

I t en

Intensity ∆Ω GHz Variation BetweMaximum and Minimum FrequencyShift Value

ensity Vector

∆Current D Variation Between Maximum and Minimum Chirp Value

k ∆ζ

al

Wave Number ps Variation Between Maximum and Minimum TemporShift Value

LD m TOD Dispersion Length

εN m C

Free Space Permittivity

Induced Magnetic ps Polarization

Temporal Shift

M m ζ psft

A, t Mode-Locker Ter Steady-State Temporal Shi

N Number of Light Modes

nm Wavelength λn

Wavelength Core Refractive

Index nm Zero-Dispersion

n Index

mCladding Refractive Micro-Meter

Group RefractiveIndex

m kg C Permeability

Free Space

m/s Group Velocity Charge Density

ation

ps Induced Electric Polariz

Pulse Width

PT τ Transmitted Power ps Lifetime of Upper Level

the

XI

Symbol Meaning

mbol Meaning

Sy

P mW τ lse Width

Launched Power ps Maximum Pu

Polarization

τ lse Nonlinear ps Steady-State PuWidth

Linear Polarization φ rad Phase Shift

P mW Average Power ibility

Optical Suscept

P mW Power χ ibility

Saturation Electric Suscept

Q - factor Ω hift Quality Factor GHz Frequency S

q Ω GHz tate t

Chirp Factor Steady-SFrequency Fhif

q ω� ncy Steady-State Chirp Optical Freque

RT Roundtrip ω

Pulse Spectrum Center Frequency

RT Roundtrip Steady-State Resonance

Frequency RT

RT=4000 Frequency = 2π Roundtrip at ω Modulation

(F ) T s Propagation Time T ns Pulse Slot Duration

TR s ime T f s/rad

th of the Roundtrip T Spectral WidFinite Gain Bandwidth

t s

e

Window in Which the Pulses are

VDelay between the Center of the Modulation Cycland the Temporal

Viewed

Parameter Determines the Number of Modes Supported by the Fiber.

1Chapter One Ultra-Short Pulses Generation

Chapter One Ultra-Short Pulses Generation

1.1 Introduction

Ultra-short pulses are very important research field. Today short

pulsed laser systems have numerous applications in areas of fundamental

research such as for medical and industrial applications, depending on the

wavelength and pulse width.

They could be used as sources in spectroscopic tools in the

laboratory for time-resolved studies of fast nonlinear phenomena in

semiconductors, or as a source in eye-safe laser radar (LIDAR).Also they

could be used as a source for a pulsed optical-fiber gyroscope or as a seed

pulse for another laser in medical applications. Other medical applications

are eye laser surgery and dentist drills. In the industry, ultra-short lasers are

used for micro-machining and marking.

Another important application of ultra-short pulses is in high-

speed optical communication systems, where an optical short pulse source

with a high repetition rate is very important for high-bit rate. [1, 2] For

these wide applications, ultra-short pulses became very important field.

Different techniques were introduced to generate ultra-short

pulses, such as: [3, 4]

a. Q-Switching,

b. Gain -Switching,

c. Pulse compression and,

d. Mode-Locking

The main difference between these techniques is the pulse width,

where in mode-locking technique a pulse width in pico-femto seconds

range could be produced. Moreover, it is of great importance since it

generates a train of Ultra-short pulses [5, 6].


Mode-locked lasers have a number of potential applications,

depending on the wavelength and pulse width.

Laser sources could be in different types, depending on the

material used as a gain medium, such as gas lasers, solid-state laser or fiber

lasers. In addition to the continuous-wave operation, fiber lasers could be

operated in a pulsed mode (mode-locked) with an output of one or more

pulses per roundtrip time.

In fiber lasers, an optical rare-earth doped fiber is used as a gain

medium to produce laser. The produced laser could be Q-switched [3, 7] or

mode-locked either: Actively or Passively. In Active Mode-Locking

technique a modulator is used to control the cavity losses and induces

pulsed operation. [8, 9] Where in Passive Mode-Locking technique an

intensity fluctuation acts with a nonlinear medium inside the cavity is used

to modulate the cavity loss without external control. [6, 10, 11, 12]

The shortest pulses which can be obtained from active mode-

locking are in the range of a few picoseconds. This because it is limited

by the speed of the electronics used to drive the modulators. While for

passive mode-locking, a femto-seconds regime could be produced, and so,

it is the preferred technique for obtaining the shortest pulses.

However, active mode-locking, allows operation at frequencies

much higher than the fundamental repetition rate, which is important for

communications applications. [13, 1]

Passive mode-locking of fiber lasers has been achieved using

different techniques such as nonlinear amplifying loop mirrors, nonlinear

polarization rotation and semiconductor saturable absorbers mirror

SESAM. [6]


1.2 Ultra-short Pulses Generation Techniques

In the last decade a rapid advances have been made, especially

for which the optical pulses from various lasers have been reduced to the

femto-second region, and the pulse peak power has increased up to multi-

terawatt level. [4]

As mentioned in previous section, numerous methods have been

developed to generate ultra-short laser pulses, among them there are two

common basic methods: Q-Switching and Mode-locking.

In this section, a brief description for the basic principles of

these two techniques, while the mode-locking technique will be

investigated deeply in next chapter, which is the aim of this thesis.

1.2.1 Q -Switching

When the lifetime of the upper laser level is much longer than the

desired pulse length of the laser output pulse, the laser medium can act as

an energy storage medium. [3, 7]

In this situation the upper laser level is able to integrate the

power supplied by the pump source. The stored energy can be released in a

short output pulse using the method of Q-switching to generate pulses with

high peak power in range of megawatts and nanoseconds regime and

repetition rates of up to several MHz [14]

In Q-switching technique, the active medium (gas,

semiconductor or doped fiber) is pumped, while the feedback from the

resonator is prevented, that is (low value of quality factor (Q) of the

resonator). The quality factor, Q is defined as the ratio of the energy stored

in the cavity to the energy loss per cycle. [3]

Q -factor = E

E …………………………… (1.1)


Equation (1.1) shows that the higher of the quality factor, gives

the lower losses of energy, which means that a large amount of energy is

stored in the gain medium, and hence a population inversion occurs.

Although the energy stored and the gain in the active medium are

high, the cavity losses are also high, so lasing action is prohibited, and the

population inversion reaches a level far above the threshold for normal

lasing action. [3]

As the amount of energy reaches a maximum level because of

losses by spontaneous emission, the Q-switch to a high Q value, suddenly

allows feedback and the intensity of light in the resonator builds up very

quickly because of stimulated emission.

Figure1.1: Generation steps of a Q-switched laser pulse.[3]

In Fig. (1.1), the steps of energy building in Q-switched

technique are shown, also it shows that the low Q of the cavity, cause the

lasing action to disable.


At the end of the flash lamp pulse, and when the inversion has

reached its peak value, the Q-factor of the resonator is switched to some

high value, which cause a photon flux starts to build up in the cavity, and a

Q-switch pulse is emitted.

Figure (1.1), shows that the emission of the Q-switched laser

pulse does not occur until an appreciable delay, during that delay , the

radiation density builds up exponentially from noise.[3]

The time for which the energy may be stored "t" is in the order of

( τ ), the lifetime of the upper level of the laser transition or a little bit less,

i.e. if t , most of the pumped energy, instead of being accumulated as

inversion energy, is wasted though spontaneous decay. [14]

τ

Many techniques are used to implement Q-Switching such as by

mechanical or electro-optical switch, where they act as a shutter inserted

inside the cavity.[14]

Mechanical Q-switches have been designed based upon rotation, so, they

inhibit laser action during the pumped cycle by blocking the light path,

causing a mirror misalignment, or reducing the reflectivity of one of the

resonator mirrors.

1.2.2 Mode-Locking

Mode-locking technique is the most important method to

generate Ultra-short pulses, where a numerous theoretical and experimental

works have been done on this field since the invention of the laser in early

of sixties of the last century.

Mode-locking of a laser is a technique, which refers to locking of

the phase relations between many neighboring longitudinal modes of the

laser cavity. A periodic train of laser pulses is produced with high peak

value and short pulse width (typically in the pico or femto seconds

regime). [15, 16]


Locking of such phase relations enables a periodic variation in

the laser output which is stable over time, and with a periodicity given by

the roundtrip time of the cavity.[10,17]

When sufficiently many longitudinal modes are locked together

with only small phase differences between the individual modes, it results

in a Ultra-short pulse which may have a significantly larger peak power

than the average power of the laser. [10]

The limitation on this technique is the length of the cavity, which

determines the pulse length. For solid-state lasers, with mode-locking

technique, Ultra-short pulses with pulse widths in the picoseconds or

femto-second regime could be obtained, due to very short length of active

medium.

In mode-locking technique, the pulse width is inversely related to

the bandwidth of the laser emission. The strong random fluctuations of the

laser output originate from the interference of longitudinal resonator modes

with random phase relations. These random fluctuations can be

transformed into a powerful well-defined single pulse circulating in the

laser resonator as shown in Fig.(1.2) , by either introducing a suitable

nonlinearity, or by an externally driven optical modulators. [3]

In the first case with suitable nonlinearity, the laser is referred

to: as passive mode-locking. This because the radiation itself, in

combination with the passive nonlinear element, generates a periodic

modulation that leads to a fixed phase relationship of the axial modes.

While in the second case with an externally driven optical modulators, it is

referred to: as active mode-locking, because an RF signal is applied to a

modulator provides a phase or frequency modulation, which leads to mode-

locking.[10,17]


Fig.1.2: Ideally mode-locked laser signal structure. (Left) A train of mode-locked

pulses (Right ) [3,6]

1.3 Historical Background

The first mode locking laser was demonstrated by Gürs and

Müller in 1963 on Ruby laser using an internal modulator driven by an

external periodic wave. The mechanism of mode locking was first clearly

explained by DiDomenico and Yariv, which is referred later as active

mode-locking.[10]

Since mode-locking technique and hence generating Ultra-short

pulses with high repetition, found wide area of applications, this made the

scientists to investigate this field deeply.[18,19,20,21]

In this section, a brief literature survey will be concerned to

explore efforts and studies that have been done to understand and

investigate mode- locking technique.

Geister and Ulrich reported in 1988 the first FM mode-locked

fiber laser. [22] Tamura, Haus, and Ippen constructed in 1992 the first all-

fiber, unidirectional, mode-locked ring laser using the non-linear

polarization rotation mechanism, NLPR .In the same year, 5-ps pulses were

demonstrated in another fiber laser mode locked at a higher repetition rate,

20 GHz.[23]

Longhi et al. demonstrated in 1994 a 2.5-GHz FM mode locked

laser with a pulse width of 9.6 ps. It is interesting to note that the pulses

obtained from FM mode-locked lasers are generally shorter than those

obtained from AM mode-locked lasers.[24]


Using rational harmonic mode locking, Yoshida and Nakazawa

demonstrated in 1996 one of the highest repetition rates to-date with a FM

mode-locked erbium doped fiber laser operating at 80–200 GHz. However,

this method was suffering from pulse-train non-uniformity, jitter, and

stability.[25] Kartner et al. analytically tried in 1995 to include not only the

effect of dispersion and nonlinearity in an AM mode-locked laser, but the

effect of the modulator as well.[26]

Grudinin and Gray used in 1997 a semiconductor saturable

absorber mirror (SESAM) to help increase the repetition rate to (> 2 GHz);

this laser operated at its 369th harmonic.[27] Abedin and co-workers

investigated in 1999 a higher-order FM mode locking in a series of papers

where they demonstrated 800-fs transform-limited pulse trains at repetition

rates as high as 154 GHz. [28]

Yu et al. used in 2000 an FM modulator in conjunction with fiber

nonlinearity to produce 500-fs pulses at a 1-GHz repetition rate. [29]

Other groups had demonstrated FM mode-locking at 40 GHz with pulse

widths of 1.37 ps and 850 fs. [30, 31] AM mode-locked fiber lasers, have

been reported in 2005 to operate at repetition rates higher

than 80 GHz. [32] The current fiber laser record of 33 fs is held by an

ytterbium fiber laser built by Buckley, Clark, and Wise, in 2006. [33]

1.4 Aim of This Thesis

The aim of this thesis is:

1. To study the effect of both normal and anomalous dispersion regimes on

output pulses in FM-Harmonic Mode-locking Ytterbium Doped Fiber

Laser.

2. Investigate the modulation frequency effect on pulse parameters

evolution, and their steady-state values.


1.5 Thesis Layout

Chapter one, demonstrates the importance of Ultra-short pulses

and basic ways to generate them.

In chapter two, a general description of optical fiber would be

investigated using of optical fiber as gain medium after doping it with

one of rare-earth atoms such as Yb or Er , and the characteristics of

ytterbium doped fiber are also shown .

Also, the parameters that effect on pulse propagation inside doped

fiber such as losses, dispersion, and nonlinearity are investigated. Both

mod-locked techniques are also investigated with focusing on active type.

In chapter three, theory and equations that govern pulse

prorogation through optical fiber starting from Maxwell's equations and

based on Non-Linear Schrödinger Equation NLSE, are demonstrated.

Mode-locked fiber laser master equation is then introduced after setting the

basic equations and assumptions. Based on master equation and modified

Ginzburg-Landau equation, pulse parameters evolution equations are

introduced using moment method.

In chapter four, a numerical simulation is done to solve the

evolution equations using fourth-fifth order Runge-Kutta method for FM-

harmonically mode-locked ytterbium fiber laser .Also discussion and

analysis of results are done.

In chapter five, conclusions and suggested for future work are

presented.

10Chapter Two Mode-Locking Fiber Lasers

Chapter Two Mode-Locking Fiber Lasers

2.1 Introduction Optical fiber is commonly used as a transmission medium in

communications where often referred to as a passive fiber. When this

fiber is doped with one of rare-earth atoms such as erbium or ytterbium

and pumped by optical source, it becomes active and could be used as a

laser source or optical amplifier. So, fiber lasers could be defined simply

as lasers with active medium is one of the rare-earth doped fibers.

An optical fiber is a circular waveguide that takes a form of a

long, thin strand of glass with a diameter of a human hair. As shown in

Fig (2.1), this fiber contains two concentric glass regions with slightly

different refractive indices. [34, 35, 36]

Refractive index is defined as the ratio of the speed of light in a

vacuum to its speed in the glass fiber medium, as in the following

relation:

Refractive Index =

………………….. (2.1)

The center of the fiber through which most of the light travels is

called the fiber core. The outer region, having a lower refractive index

than the inner region, and is called the cladding. Such fibers are generally

referred to as step-index fibers to distinguish them from graded-index

fibers as shown in Fig. (2.2) ,where the refractive index of the core

decreases gradually from center to core boundary. [36]


Fig 2.1: Optical fiber composition. [36]

Typical values for core and cladding diameters in standard

glass optical fiber are 50 m and 125 m respectively. A surrounding

plastic coating is normally applied to protect the glass fiber. Two basic

optical fiber types exist: single-mode and multi-mode, as shown in

Fig (2.3). The main difference between them is the dimension of the fiber

core. A single-mode fiber typically has a core diameter of 10 m, which

allows only one mode of light at any time to propagate through the

core. [36]

Fig. 2.2: Step and Graded index optical fiber.


Multi-mode fiber has a much larger core (normally a 50 m or

62.5 m diameter), allowing hundreds of modes of light to move through

the fiber simultaneously as shown in Fig.(2.3).

Fig. 2.3 Optical fiber modes. [35]

The optical fibers configuration allows for total internal

reflection of light at the boundary between core and cladding. Light

reflects (bounces back) or refracts (changes its direction while penetrating

a different medium), depending on the angle at which it strikes the

core/cladding boundary.

The light waves are guided to the other end of the fiber due to

continuously reflected within the core, as shown in Fig. (2.4). In this way,

the fiber core acts as a waveguide for the transmitted light. Controlling the

angle at which the light waves are transmitted makes it possible to control

how efficiently they reach their destination. As a result, the composition

of the cladding glass relative to the core glass determines the fiber's ability

to transmit light. [36]


Fig. 2.4 Light propagation through optical fiber

Because the cladding absorbs or scatters only negligible light

from the core, the light wave can travel great distances. However, when

light propagates in the core of an optical fiber, some small part of the light

is lost due to scattering phenomena.

The extent that the signal degrades depends on the purity of the

glass and the wavelength of the transmitted light. This loss or attenuation

is normally quoted in a logarithmic loss per unit fiber length, i.e. in dB per

fiber length. At wavelengths typically used in remote sensing systems,

attenuation in a single-mode fiber is about 0.2 dB/km. The attenuation in a

multi-mode fiber is somewhat higher.

There are two parameters which characterize an optical

fiber.[34]

1. The relative core-cladding index difference

∆ ………………………………………………. (2.2)

Where,

n

n n /

k 2π/λ

a: the core radius, and

: core refractive index,

n : cladding refractive index.

2. The V parameter defined as:

V = k ……………………………………………. (2.3)

Where

,


λ�: the light wavelength.

determines the number of modes supported by

the fiber.

s mentioned earlier, the main difference between the single-

mode and

sers are very closely related to fiber amplifiers, since

they are l

rth doped fiber is used as the gain medium

(cavity) o

th light of a wavelength appropriate to

the lasing

The V parameter

If V 2.405, then the step index fibers is referred to be single

mode fiber.

A

multimode fibers is the core size, so the later is typically

25–50 μm for multimode fibers. For the outer radius b, the numerical

value is less critical as long as it is large enough to confine the fiber

modes entirely. A value of b= 62.5 m� is commonly used as standard for

both single-mode and multimode fibers. [34]

2.2 Fiber Laser

Fiber la

asers with active medium is made of rare earth doped fiber, as

mentioned earlier. [6, 36, 37]

A section of rare-ea

f the laser, while the mirrors can be made in various ways but

the use of Fiber Bragg Gratings (FBGs) is very attractive because of their

wavelength-selective nature. [35]

The laser is pumped wi

medium, (as an example: 980 nm or 1480 nm for erbium).

Figure (2.5) shows, a simple example of a fiber laser which is constructed

from two FBGs and a length of erbium doped fiber. It is an optical

amplifier with mirrors on the ends of the fiber to form a cavity.


Fig. (2.5): Fiber laser using FBGs. According to design requirements, exit mirror reflectivity could be between 5% and 80% at certain wavelength.[35]

Pumped laser of 980 nm wavelength enters the cavity through

the left-hand FBG. Both FBGs are resonant (reflective) at a very specific

chosen wavelength in the 1550 band , so the 980 nm light will pass

straight through the FBG without attenuation.[6] Due to optical pumping,

atoms will be excited to higher levels.

Consequently a spontaneous emission will begin in the erbium

doped fiber very quickly. Since spontaneous emission is random in

direction, so most of it will not be in the guided mode and will leave the

cavity quite quickly.

Also, it is random in wavelengths, so will not be at exactly the

right wavelength to be reflected by the FBGs and will pass out of the

cavity straight through the FBG mirrors. But some spontaneous emission

will (by chance) have exactly the right wavelength and will happen to be

in the guided mode.

In this case, lasing will begin, since these emissions will be

reflected by the FBGs and amplified in the cavity. [35]


2.2.1Factors Affect on Fiber Laser Characteristics

There are some factors which have effect on fiber laser

characteristics such as:

1. Lasers could be produced in different wavelength bands,

depending on rare earth dopants. [6, 38, 39]

NdNdPrErYb

Also the level of rare earth dopant used in the glass is another

factor which affect on fiber laser. Some glass hosts cannot be doped to

higher concentrations in “ZBLAN” glasses. [35]

2. Another factor affects on operational characteristics of the

lasing medium , it is the type and composition of glass used in the “host”

fiber. This is because the host plays a part in the energy state transitions

necessary to support stimulated emission.

2.2.2 Rare-Earth Doped Fibers significant properties

Rare-earth doped fibers are attractive as laser gain media due to

the following significant properties: [6, 7, 35, 40]

1. High power output (in hundreds watts or even several kilowatts), due to

2. Broad gain bandwidth

at 0.9 μm at 1.08 μm at 1.06 μm at 1.55 μm at 1055 μm

very high concentration ratio. As example, erbium in silica glass can only

be used to a maximum of about 1% but it can be used at significantly

3. The used grating characteristics determine the exact output

wavelength.

high gain and high efficiency.


3. Excellent beam quality

4. Directly pumped by laser diodes

line-width, a 10 kHz line-width has been produced.

gths, since FBGs can be manufactured to very

e properties make doping optical fibers , which are fiber

lasers sou

.2.3 Disadvantages of Mode-locked Fiber Laser Systems

em superior

to classic

could affect fiber laser operation and hence its

output cha

ible, it should preferably

be implem

imitations in fiber

lasers an

5. Low noise

6. Tunability

7. Very narrow

8. Good soliton generation

9. External modulation

10. Preselected wavelen

accurate wavelength tolerances

11. Low cost

Thes

rces ,very important components in modern communication

systems (for their ability to generate transform-limited pulses [41] ) and ,

hence attractive as a gain medium in mode-locked lasers.[42]

2

Although fiber lasers have many features, making th

al solid-state lasers, they have many disadvantages need to

overcome them, such as:

1. Environment

racteristics, such as changing temperatures, air convection etc.,

so, fiber laser should be environmentally stable.

2. To make fiber laser as stable as poss

ented without sections of free space optics.

3. Nonlinearities, are another important l

d amplifiers which affecting the pulse due to the tight

confinement and long interaction lengths in optical fibers.


4. Solid-state lasers often require maintenance, due to high

power consumption by the system.

The potential of making compact, rugged laser systems with

low power consumption at relative low price make amplified fiber lasers a

very promising alternative to classical solid state lasers.

In the following section, a description of the energy diagram

and the transitions of ytterbium doped fibers Yb will be shown.

2.2.4 Ytterbium Doped Fibers The significant features of ytterbium-doped fiber made it

attractive as optical amplifier and laser source. It has broad-gain

bandwidth, high efficiency and broad absorption band [43].

It has high quantum efficiency (~ 95 %), since lasing band is

very close to the pump wavelength. [38] Very high doping concentrations

are possible in ytterbium doped fibers, enabling very high single pass

gains and high slope efficiencies of up to ~ 80 %.

Ytterbium absorption band extends from below (850 nm) to

(980 nm) and from (1010 nm) to above (1070 nm), and has peak

absorption at 976 nm as shown for the emission and absorption spectrum

as shown in Fig.(2.6).Therefore they can generate many wavelengths of

general interest, e.g. for spectroscopy and pumping other fiber

lasers.[6,37].


Fig.2.6 Absorption (solid) and emission (dotted) cross sections of Yb . [35]

Yb

Yb

Spectroscopy is very simple in comparison with other

rare-earth doped fibers .It is considered as two main level systems (ground

and excited levels) with other sublevels as shown in Fig. (2.7). [37]

When Yb doped fiber is typically pumped into the higher

sublevels, it behaves as a true three-level system for wavelengths below

about (990 nm), as shown in Fig. (2.7). While at the longer wavelengths,

from (~1000 to ~1200 nm), it behaves as a quasi-four-level system.

Fig. 2.7: Energy level of Ytterbium doped silica fibers. [35]

Doped fibers are very efficient sources. The emission

wavelength can be more selected by a careful choice of the pump

wavelength. The gain of an ytterbium doped fiber can be well modeled by


a Gaussian function with FWHM of (~ 40 nm) and with a central

wavelength at (1030 nm).

In the present time, Y with (1020 nm) being for pumping

praseodymium doped fluoride fiber amplifiers (amplification in the

1.3 μm region) and with (1140 nm) for pumping thulium doped fibers.[38]

b

2.3 Losses, Dispersion and Nonlinearity in Doped Fiber

When light enters one end of the fiber it travels (confined within

the fiber) until leaving the fiber at the other end. It will emerge

(depending on the distance) and become much weaker, lengthened in

time, and distorted.

In the following section, a brief study of these parameters which

are mainly: losses, dispersion and nonlinearity, that affect on pulse

propagated in optical fiber. [36]

2.3.1. Fiber Losses

Losses are referred to the attenuation of the pulse propagated

through the optical fiber .This attenuation is defined as the reduction in

the output signal power as it travels in distance through the optical

fiber. [34, 36]

This attenuation is due to several factors such as, material

absorption losses, Rayleigh scattering losses and bending losses which are

contributing dominantly. Other mechanisms for scattering light are a

result of the existence of inhomogeneities in the materials based on

compositional fluctuations or the presence of bubbles and strains

introduced in the process of jacketing or cabling the fiber. [4]

A brief description for each type of losses will be investigated

in the following sections.


2.3.1.1 Material Absorption

When light travels through the fiber it will be weaker because

the glass absorbs light. In fact, the glass itself does not absorb light, but

the impurities in the glass absorb light with the wavelengths of interest

(infrared region). [35] However, even a relatively small amount of

impurities can lead to significant absorption in that wavelength band. [34]

Generally the OH ions are the most important impurity affecting

fiber loss, as shown in Fig (2.8).

Fig. 2.8.Typical Fiber Infrared Absorption Spectrum. [36]

Also the figure displays losses in single and multimode fiber

and Rayleigh scattering .The lower curve shows the characteristics of a

single-mode fiber made from a glass containing about (4% )of germanium

dioxide (GeO2) dopant in the core. The upper curve is for modern graded

index multimode fiber.

Due to higher levels of dopant used, attenuation in multimode

fiber is higher than in single-mode. The dashed curve shows the

contribution resulting from Rayleigh scattering.

During the fiber-fabrication process, special precautions are

taken to ensure that the level of OH-ion is less than one part in one

hundred million. [34]


Measuring power loss is an important fiber parameter during

transmission of optical signals inside the fiber. Due to high technology

used in producing optical fiber, a significant reduction in material losses is

achieved as shown in Fig. (2.9).

Fig. 2. 9. Transmission Windows. Upper curve shows the absorption characteristics of

old fiber in the 1970s, while lower curve is for modern fiber. [35]

As shown in Fig. (2.8) and (2.9), the fiber exhibits a minimum

loss of about (0.2 dB/km) near (1.55 μm). Then losses are considerably

high at wavelengths shorter and longer than (1.55 μm), reaching a level of

a few dB/km in the visible region.

In Fig. (2.9), three windows (or bands) are shown to facilitate

losses–wavelength dependency study. These windows are: short, medium

and long wave respectively, according to their timetable development.

If P is the power launched at the input of a fiber of length l,

then the transmitted power P is given by: [34]

T

T P = P exp (-αl) ………………………………………………….. (2.4)

Where the attenuation constant (α) �is a measure of total fiber

losses from all sources. Usually (α) is expressed �in units of dB/km

using the following relation:[34]


= log 4.343

R R

on the constituents of the fiber core.

= λ μ

fibers are dominated by Rayleigh scattering. Another reason for this type

…….……………………………. (2.5)

Eq. (2.4) was used to relate and , Eq. (2.5) is

wavelength dependant.

2.3.1.2 Rayleigh Scattering

Most light loss in a modern fiber is caused by scattering .

Rayleigh scattering is a fundamental loss mechanism, caused by the

interaction of light and the granular appearance of atoms and molecules

on a microscopic scale. [4, 36]

Rayleigh scattering originates from density fluctuations frozen

into the fused silica during manufacture, resulting local fluctuations in the

refractive index, causing to scatter light in all directions. It varies as

and is dominant at short wavelengths. [34]

Since this loss is intrinsic to the fiber, it sets the ultimate limit

on fiber loss. The intrinsic loss level (shown by a dashed line in Fig. (2.8)

is estimated to be (in dB/km) as in the following relation: [34]

α =C /λ ……………………………………………………... (2.6)

Where the constant is in the range (0.7–0.9 dB/(km-μm )) depending

As 0.12–0.15 dB/km near = 1.55 m, losses in silica

of losses is the variations in the uniformity of the glass cause scattering of

the light. Both rate of light absorption and amount of scattering are

dependent on the wavelength of the light and the characteristics of the

particular glass.


2.3.1.3 Bending Losses

Another type of losses arises from bending optical fiber .At the

bend , the propagation conditions alter and the light rays which would

propagate in a straight fiber are lost in the cladding .[36]

Two classes of fiber losses arise from either large-radii bends or

small fiber curvatures with small periods, as shown in Fig.(2.10).These

bending effects are called macro-bending losses (due to tight bend ) and

micro-bending losses (due to microscopic fiber deformation, commonly

caused by poor cable design ), respectively.

It is important to characterize these losses because they are

important if there is a need to wrap fiber.

Macro-bending loss Micro-bending loss

Fig. 2.10 Bending losses in optical fiber

The critical angle , is defined as the angle of light incident

where grater than it, light will radiate away instead of reflects inside the

fiber.

As shown in Fig. (2.10), when no bending, fiber bounce angle

α < , but at the bend α > , so total internal reflection condition is not

satisfied, and hence some light leaks out into cladding.

If a fiber is bent from the straight position, the light may be

radiated away from the guide, causing optical leakage. As the radius of


curvature of the fiber bends decreases, the bending loss will increase

exponentially, so the critical radius is the bend radius which below it,

losses increase rapidly. [35, 36]

2.3.2 Dispersion

It is defined as the broadening in the optical pulse due to the

variation of refractive index with wavelength as the pulse of light spreads

out during transmission on the fiber.[10,34]

When the pulse propagates inside the fiber, each spectral

component travels independently, and hence suffers from time delay or

group delay per unit length in the direction of propagation.[36,44]

The broad bandwidth frequency components of a transform limited pulse

experience an index of refraction based on their frequency n(ω).A short

pulse becomes longer due to broadening and ultimately joins with the

pulse behind, making recovery of a reliable bit stream impossible as

shown in Fig. (2.11).

Fig. 2.11 Dispersion effect on propagated pulses.

The circles in the figure represent fiber loops. [9]

In communication systems as bit rates increase, dispersion

becomes a critical aspect and limits the available bandwidth. This because


dispersion broadening effect will make bit interval longer. Consequently

fewer bits transmuted and hence low bit rate.

There are many kinds of dispersion, each type works in a

different way, but the most important three are discussed below.

2.3.2.1 Chromatic Dispersion

A chromatic dispersion is a sum of two types of dispersions

which are: [34, 36]

a. Material dispersion

It is intrinsic to the optical fiber itself, which arises from the variation of

refractive index with wavelength. [44]

b. Waveguide dispersion

Which is a function of design of the core and cladding of the fiber, and

arises from the dependence of the fiber's properties on the wavelength.

Fig (2.12) displays the two types of dispersion and their

resultant.

Fig. 2.12: Chromatic dispersion types and their sum


2.3.2.1.1 Material Dispersion

Material dispersion could be explained as following:

The bound electrons of a dielectric (optical fiber) interact with an

electromagnetic wave propagated inside it, causing to the medium to

response, depending on the optical frequency ω. This response is called as

a material dispersion, arising from the frequency dependence of the

refractive index n (�ω).

Essentially, the origin of material dispersion is related to the

characteristic resonance frequencies at which the medium absorbs the

electromagnetic radiation through oscillations of bound electrons. [34]

Since lasers and LEDs produce a range of optical wavelengths

(a band of light) rather than a single narrow wavelength, therefore the

fiber has different refractive index characteristics at different wavelengths

and hence each wavelength will travel at a different speed in the fiber.[35]

Thus, some wavelengths arrive before others and hence, a signal pulse

disperses. [36]

As shown in Fig (2.13), a change in refractive index due to

different wavelengths spectrum .The refractive index could be well

approximated by the Sellmeier equation: [34, 44]

n ω 1 ∑B

……………………………………. (2.7)

Where is the resonance frequency and B is the strength of jth resonance.


Fig. 2.12: Refractive index n and Group index n versus wavelength for fused silica. [34]

ω n ω

Dispersion plays very important role in propagation of short

optical pulses since different spectral components associated with the

pulse travel at different speeds given by c / n(ω).

Dispersion-induced pulse broadening can be harmful for

optical communication systems, even when the nonlinear effects are not

important. [44, 45]The combination of dispersion and nonlinearity can

result in a qualitatively different behavior, when system is brought into

nonlinear regime. [21]

By expanding the mode-propagation constant β by Taylor series

about the frequency ω at which the pulse spectrum is centered, the

effects of fiber dispersion could be accounted mathematically: [34, 46, 47]

β β ω ω β β ω ω

β βω

(2.8) Where:

ω ω (m = 0, 1, 2) ………………………… (2.9)


β ed to the refractive index n and their derivatives are :

and β are relat

β n ω ……………………… (2.10)

β ω2 ……………………………… (2.11)

re:

oup velocity,

: the group index,

c: the light speed .

point of view, the envelope of an optical pulse

elocity while the parameter β represents dispersion

of the gro

Fig. 2.13: β and d as a function of wavelength for fused silica.[34]

Fig.(2.13) shows that β vanishes (becomes zero) at a

avele bo

Whe

: the gr

From physics

moves at the group v

up velocity and is responsible for pulse broadening. [34, 45]

Hence this effect is called as the group-velocity dispersion (GVD), and β

is the GVD parameter. [44]

w ngth of a ut (1.27 μm) and it is negative for longer wavelengths.


This wavelength is called to as the

the term of dispersion appears and must be considered. This

new term rd

an

both in the linear

and nonlinear regimes. Adding it, is

Fig.2.14: Dispersion in standard single-mode fiber 2.3.2.1.2 Normal and Anomalous Dispersion

Depending on the sign of the GVD parameter, nonlinear effects

optical fibers can exhibit qualitatively different behaviors. As shown

) the fiber is in so-called

normal dispersion regime as 0. In this regime, high-frequency

(blue-shifted) components of an optical pulse travel slower than low-

pone

zero-dispersion wavelength and is

referred to as .

When λ= ,it doesn't mean that dispersion becomes zero ,

however, ano r

is called thi order dispersion TOD or , which is the cubic

term in Eq.(2.8).[47]

This higher-order dispersive effects c distort ultra-short

optical pulses by asymmetrically broaden pulses [48]

necessary only when the wavelength

λ approaches to within a few nanometers. Fig.(2.14) demonstrates

normal and anomalous dispersion regime as functions of wavelength in

single-mode fiber .

in

in Fig. (2.14) when λ ,(with λ 1.3µmD

frequency (red-shifted) com nts of the same pulse.


In the anomalous dispersion regime when β 0, the opposite

occurs. Optical fibers exhibit anomalous dispersion when the light

wavelength exceeds the zero-dispersion wavelength λ λD as shown in

Fig. (2.14

tween dispersion and nonlinearity.

o Eq.(2.7) .

uction itse

and added

Figure 2.15 Dispersion parameter D as a function of wavelength for three types of

fibers. Single-clad SC, double-clad DC, and quadruple-clad fibers QC.[34]

). [46,47]

Anomalous-dispersion regime is very important in study of

nonlinear effects since in this regime, the optical fibers support solitons

through a balance be

Fig. (2.14) is considered the bulk-fused silica, while the

dispersive behavior of actual glass fibers deviates from that shown in

these figures due to following two reasons :

1.In fiber laser ,where fiber is used as gain medium , the fiber

core may have small amounts of rare earth dopants such as (Er , Yb )

in this case , dopants effect should be added t

2. Due to dielectric wave-guiding, the effective mode index is

slightly lower than the material index n(ω) of the core, red lf

being ω dependent. Hence, a waveguide contribution must be considered

to the material contribution to calculate the total dispersion.


2.3.2.1.3 Waveguide Dispersion

In spite of a ve

which is caused by the shape and index

be controlled by careful design .In f

to counteract material di

Waveguide effect on

dispersion wavelength λD

her Deasured total dispersion of

commonly

ry complex effect of a waveguide dispersion

profile of the fiber core, this can

act; waveguide dispersion can be used

spersion. [34, 36, 44]

β is relatively small except near the zero-

where the two become comparable. The

w

the

a single mode fiber.

s the dispersion parameter D that is

aveguide effect is mainly to shift λD slightly toward longer wavelengths,

e ( λ ~1.31 μm) for standard fibers. Figure (2.15) shows w

m

The quantity plotted i

used in the fiber-optics instead of β . It is related to β by the

relation: [34]

D β …………………………………. (2.13)

Different type of attenuation in fiber and dispersion are plotted

ig. 2.16 :Left :attenutation versuse wavelength . Right :attentuation and dispersion versuse wavelegth.

as a function of wavelength as shown in Fig. (2.16).

F


2.3.2.1.4

ic but it

contains im r is changed in

odes travel

at the same speed, and the polarization modes are said to be degenerated.

While in the case where the two modes travel at different

speeds, the fiber can be described as birefringent. [36]

Over a length of the fiber , the polarization states travel at

different speeds, so the states will be unsynchronized , which results that

the signal energy reaches the fiber end at different point in time , and

hence pulse spreading or dispersion arises [49] as shown in Fig ( 2.17) .

Polarization mode dispersion PMD is directly proportional with

square root of distance. When operating fibers near zero chromatic

dispersion, PMD effects become crucial.[34]

2.3.3 Non-Linear Effects in Optical Fibers The previous explanation of effects are considered as power

dependent (i.e. m and now, effects

behavior) that are power dependent will be considered. Such behavior

P

ly symmetr

perfections, so light travelling down such a fibe

polarization. When light transmitted in a single mode fiber, it travels in

two orthogonal polarization modes. In a perfect fiber, both m

olarization Mode Dispersion (PMD)

Conventional optical fiber is cylindrical

Fig. 2.17:Polarization mode dispersion

in ainly depend upon wavelength),

(


depends o

nto two types as shown

Fig. (2.18):

BS)

elengths propagate simultaneously inside a fiber. As

en them, new waves are generated.

Cross-pha nied by self-phase modulation

(SPM) an dex seen by an optical

n the intensity of other copropagating beams.

tral broadening of co-

prop re of XPM is that, for

equally in

e of any dielectric

becomes n

n the power (intensity) of light propagating inside the fiber is

called non-linear optical effect, which includes the following

phenomena:[34,50]

1. Nonlinear Refraction effects:

a. Self Phase Modulation (SPM)

b. Cross phase modulation effects. XPM

2. Stimulated Scattering, is subdivided i

in

a. Stimulated Brillouin Scattering (S

b. Stimulated Raman Scattering (SRS)

SPM, is a phenomena occurs as a result of intensity dependence

of refractive index in nonlinear optical media which leads to spectral

broadening of optical pulses, and hence distortion. [45]

XPM is a phenomenon occurs when two or more optical fields

having different wav

a result of interaction betwe

se modulation is always accompa

d occurs, since the effective refractive in

beam in a nonlinear medium depends not only on the intensity of that

beam but also o

XPM is responsible for asymmetric spec

agating optical pulses. An important featu

tense optical fields of different wavelengths, the contribution of

XPM to the nonlinear phase shift is twice that of SPM. [50, 51, 52]

When the light intensity increases, the respons

onlinear for intense electromagnetic fields, and so optical fibers

do. [34, 35] Stimulated inelastic scattering is a nonlinear effect results

from the optical field transfers part of its energy to the nonlinear medium.


While, elastic effect is for no energy exchanged between the

electromagnetic field and the dielectric medium.

Increasing intensity above a threshold causes a stimulated

scattering which is defined as a transferring energy from the incident

wave to a

oustic

phonons

mentioned in chapter one, mode-locking of a laser refers to a

cking of multiple axial modes in a laser cavity by enforcing coherence

between the phases of different modes. This done by a relatively weak

modulation synchronous with the roundtrip time of radiation circulating in

wave at lower frequency (longer wavelength) with the small

energy difference being released in the form of phonons.

SBS and SRS as an inelastic scattering, the main difference

between them is that: optical phonons participate in SRS while ac

participate in SBS. In a simple quantum-mechanical picture

applicable to both SRS and SBS, a photon of the incident field (called the

pump) is annihilated to create a photon at a lower frequency and a phonon

with the right energy and momentum to conserve the energy and the

momentum.

Fig. 2.18 Stimulated Scattering

Even though SRS and SBS are very similar in their origin,

different dispersion relations for acoustic and optical phonons.

2.4 Mode-Locked Lasers

As

lo


the laser, a pulse is initiated and can be made shorter on every pass

through the resonator.[10] It results in a short pulse which may have a

significantly larger peak power than the average power of the laser. The

shortening process is limited by finite bandwidth of the gain. [10, 17]

The origin of mode-locking is best understood in the time

omain. A laser in stea m, where the gain per

roundtrip i

a

ser may favor a superposition of

power.[53]

2.4.1 Physics of Mode-Lock

d dy state is a feedback syste

s balanced by the losses.

of optical power) element is introduced into the cavity, which introduces

If a nonlinear (i.e. nonlinear in terms

higher loss at lower powers, the la

longitudinal modes corresponding to a short pulse with high peak

However, a further requirement for obtaining stable mode-

locking is that the pulse reproduces itself after one roundtrip (within a

total phase shift on all the longitudinal modes). [20, 54] The phase

relations between different modes are affected by many factors such as

dispersion, gain bandwidth, nonlinear phase shifts etc. (where these will

be investigated when modeling mode locking equations). [55]

ing

Lasers output is not monochromatic; rather, they usually operate

simultaneously in a large number of longitudinal modes falling within the

gain bandwidth as shown in Fig. (2.19). [6, 20, 53 ]

Mode in laser Cavity Laser Spectrum

Fig.2.19 Laser longitudinal modes and gain bandwidth.


For a longitudinal mode to be supported, Eq. (2.21) must be

satisfied:

2L/λ = N ……………………………………………………….. (2.21)

L: cavity

The frequency spacing among the modes is given by Eq. (2.22),

∆ c 2nL …………………………………………………... (2.22)

Where:

c: light speed

n: cavity refractive index , and the product ( nL ) represents the optical

path .[10,4

ode operation is due to a wide gain bandwidth compared

As shown in Fig ( 2. 19), if the gain bandwidth is broader than

ore than one longitudinal mode can oscillate. If a

odes with equal amplitude,( E ), each has a

φ , then the total amplitude can be expressed by

]

E t E ∑N

Hence intensity for N modes will equal to N times the intensity

ode as in following equation:

Where:

length,

N: integer number and,

λ : laser wavelength.

1]

Multim

with a relatively small mode spacing of fiber lasers ∆ ~10 MHz .

this mode spacing, m

cavity containing N light m

frequency ω and a phase

the following relation: [4,56

e ………………………………………(2.23)

of one m


I t |E t | E ∑ eN NE ………………….. (2.24)

to

each other, and the process is called mode locking. [57]

. These sidebands overlap with the

neighboring modes when F ∆ , where F represents the

odulation frequency.

ynchronization. [6]

φ into Eq. (2.23), the combined

field amplitude can be rewritten as in Eq.( 2.25)

E tN ∆

When a constant linear relationship is considered between the

phases of the laser modes, then all oscillating modes are phase-locked

To create such phase locking, an intra-cavity loss or gain

modulator operating synchronously with the cavity roundtrip frequency

∆ is required. [58]The modulation effect is to generate sidebands.

Consequently, these sidebands will give rise to an energy transport

between neighboring modes

m Such an overlap leads to phase

s

Adding the linear phase relationship, e.g. as a constant phase

offset from mode to mode, α φ

:[56]

E e N N ∆ ………………………… (2.25)

Hence, intensity becomes:

I t EN ∆

∆ EN ∆

∆ ………… (2.26)

re ∆ω ω ω , and, α φ φ

A schematic waveform could be drawn for resultant locked

phases using Eq. (2.23) and Eq. (2.26) and by taking, e.g. and

0 ,as shown in Fig. (2.20

Whe

).


In this figure, the upper part demonstrate the real part of the

complex field of five laser modes as function of time (green), where the

lower par

and 50

de with continuous

ave (cw) output (red).

Fig. , and pulses generated from phase matching ower ).[56]

The fives modes demonstrate an equal phase once per roundtrip

ount of energy as the continuous output per

roundtrip, but concentrated in a small time window.

More modes have a shorter time of coincidence, which results

in narrower peaks with larger peak intensities.

Examining Eq. (2.26), it is clear that the maximum intensity is

increased by a factor of N over the average intensity.

t demonstrate the intensity as function of time for two different

numbers of modes, 5 (magenta) (blue), and the normalized

intensity in case the laser is operated in a single mo

w

2.20: The locked modes (upper)(l

∆ 1/∆ , whereas at all other times the modes interfere

destructively.

Also, once per roundtrip, a pulse with large intensity occurs,

containing the same am


I t N E ……………………………………………….. (2.27) Where:

N: the number of longitudinal modes,

E : the amplitude of longitudinal modes. 2.4.2 Parameters Limiting Pulse Duration

There are many parameters that govern pulse width among them

are: the optical cavity length, the optical path (nL) and the number of

oscillating modes, N.

Generating shortest pulse duration is related to these parameters

s in the f

L

N

a ollowing relation: [56]

∆t ,π

∆ωN ∆ N ………... (2.28)

Where,

odes means wide

bandwidth

2.4.3 Tim

nverse of the gain

andwidth, it is often referred to as bandwidth-limited pulses. Also could

e spectrum. [59]

pulses with a Gaussian temporal

sh e following relation:

∆ N gainbandwidth.

So, large number of longitudinal m

, hence shorter pulse width. [6]

e-Bandwidth Product

When pulse duration is equal to the i

b

be defined as the shortest pulses that can be obtained from a given

amplitud

As an example, in case of

ape, the minimum pulse duration as in th


∆t ,.

N∆ …………………………………………….. (2.29)

The value (0.441) is known as the time-bandwidth product and

depends on the pulse shape. The minimum attainable time-bandwidth

roduct of a hyperbolic secant squared pulse shape is (0.315) and (0.11)

for a single sided exponential shape.

de-Locking Techniques

locking technique is divided into two types,

Active an

e that are able to initiate mode-

cking by using an optical effect in a material without any time varying

ockers, on the other hand, are those that

use some

ssive to get benefit of the advantages of both

types .

ctive Mode-Locking

Active mode locking is a technique based on active modulation

chieved, by incorporating optical modulator inside the laser cavity such

as an aco

p

2.4.3 Mo

Generally, mode

d Passive mode locking. The difference between them is very

simple.[10] Passive mode-locking are thos

lo

intervention. While active mode-l

of externally modulated media or device.

Sometimes another type of mode locking is referred to as

hybrid mode locking is used , which is in fact a combination of the two

main types active and pa

2.4.3.1 A

of the intracavity losses or the roundtrip phase change. This can be

a

usto-optic or electro-optic modulator, such as Mach-Zehnder

integrated-optic modulators, or a semiconductor electro absorption

modulator. [10, 17]


Active mode locking requires modulation of either the

tion) or the phase of the intracavity

optical fi

se with the "correct"

timing can

ortening is offset by other effects (e.g. the finite gain bandwidth)

which tend to broaden the pulse.

g,

operation the roundtrip time of the cavity must quite precisely match the

amplitude AM (amplitude modula

eld, FM (frequency modulation) [60] mode locking at a

frequency F equal to (or a multiple of) the mode spacing, ∆ . [6, 17]

2.4.3.1.1 AM Mode-Locking

The principle of AM active mode locking by modulating the

cavity losses is easy to understand.[17,61] A pul

pass the modulator at times where the losses are at a minimum.

Still, the wings of the pulse experience a little attenuation, which

effectively leads to (slight) pulse shortening in each roundtrip, until this

pulse sh

In simple cases, the pulse duration achieved in the steady state

can be calculated with the Kuizenga-Siegman theory. It is typically in

the picoseconds range and is only weakly dependent on parameters like

the strength of the modulator signal.[57]

2.4.3.1.2 FM Mode-Locking

Active mode-locking also works with a periodic phase

modulation (instead of amplitude modulation), even though this leads to

chirped pulses.[ 2,8, 11,59] This technique is called frequency modulation

FM mode-locking .In both types of active mod-lockin for stable

period of the modulator signal.


A significant frequency mismatch between laser cavity and

drive signal can lead to strong timing jitter or even chaotic behavior.

FM mode-locking involves the periodic modulation of the

roundtrip phase change [62, 60], is achieved using an electro-optic

modulator, such as Mach-Zehnder integrated-optic modulator, which

often are

arated from each other so that they

are uncoupled.[48,57]

r Interferometric (MZI) Modulator

of

a nonlinear crystal by an electric field in proportion to the field strength.

When an

generate ultra-short pulses, usually with pico-second pulse durations. In

used as phase modulators. As shown in Fig. (3.1), MZI is an

electro-optic modulator (EOM) device, which consists of two symmetric

Y-branches. These branches are connected back to back by two parallel

channel waveguides that are well sep

Fig. 2.21: Mach-Zehnde

They are based on the linear electro-optic effect (also

called Pockels effect), i.e., the modification of the refractive index

electric field is applied to the crystal via electrodes, refraction

index will change linearly with the strength of an externally applied

electric field E.

As a result changes in the phase delay of a laser beam will take

place. MZI allows controlling the power, phase or polarization of a laser

beam with an electrical control signal. FM mode-locking is capable to


most cases, the achieved pulse duration is governed by a balance of pulse

shortening through the modulator and pulse broadening via other effects,

such as the limited gain bandwidth.[13]

2.4.3.2 Passive Mode-Locking

Passive mode locking is an all-optical nonlinear technique

which produces ultra-short optical pulses, without requiring any active

component (such as a modulator) inside the laser cavity. [11, 19, 63]

Instead a saturable absorber or a Kerr lens as an example is used inside

the cavity. intensity-

ependen

oss than the central part, which is

sorber. The net result is that the pulse is

shortened

It is called passive mode-locking since these

d t shutters transmit light when the intensity is high, such that they

do not need an external control because they are controlled by the arrival

time of the pulse itself. The most important type of absorber for passive

mode locking is the semiconductor saturable absorber mirror, called

SESAM.[11]

The nonlinear effect of saturable absorbers enables ultra-short

pulses in the femto-second regime, providing that the gain profile is

sufficiently large. Passive mod-locking basic mechanism could be

explained easily as follows:

The absorption of the fast saturable absorber, SA, can change

on a timescale of the pulse width. During pulse propagation through such

an absorber, its wings will suffer more l

intense enough to saturate the ab

during its journey through the absorber. [6]

2.4.3.3 Hybrid Mode-Locking

Hybrid mode locking is a combination of active and passive

mode locking, where both an RF signal and a passive medium are used to


produce ultra-short pulses. It takes advantage of active mode-locked

stability and the saturable absorber’s pulse shortening mechanisms.

45Chapter Three The Model of Mode-Locked Fiber Laser

Chapter Three

The Model of Mode-Locked Fiber Laser

3.1 Pulse Propagation in Optical Fibers

To describe the mode-locked fiber lasers and hence setting the

master equation that governs this technique, it is necessary to consider the

theory of electromagnetic wave propagation in dispersive, nonlinear

media. [34, 36]. Pulse propagation in optical fibers is governed by the

Nonlinear Schrodinger Equation (NLSE), which must generally be solved

numerically since it has no analytic solution. [45, 64]

The purpose of this chapter is to obtain the basic equation that

satisfies propagation of optical pulses in single-mode fibers. Then the

equations that concern the evolution of pulse parameters during each

roundtrip will be introduced. These equations will be solved numerically

using fourth-fifth order Runge-Kutta .

Since deriving mode-locked master equation and all details

related are beyond the scope of this thesis, the general procedures and steps

will be shown to give an idea of how to implement these derivations and all

assumptions and approximations which are used.

3.1.1 Maxwell Equations

Starting with Maxwell four equations, which are very well-

known equations that govern the propagation of electromagnetic waves.

Electromagnetic waves consist of two orthogonal components of electric

field vector and magnetic field vector . [34, 36, 53]


Maxwell equations are:

1. …………………………………………………….(3.1)

2. ………………………………………….……… (3.2)

3. · ……………………………………………………….(3.3)

0

: the electric flux density ,

d magnetic fields and

4. · ...……………………………………………………..(3.4) Where:

: the current density vector ,

: the charge density ,

: the magnetic flux density .

Due to the electric an propagation

inside the itimedium, an electrical and magnetic flux dens es and

respectively arise. They are related to and through the following

equations: [34, 36]

=ε ………………………………………………………... (3.5)

= μ + …………………………………………………………. (3.6)

W ere:

ittivity of free space,

: induced electric polarization,

: induced magnetic polarization, and

h

ε : perm

: permeability of free space ,


ε = 1/ ,

le magnetic field propagation is governed by Eqs. (3.1) to

uld be taken to the effects of these

equations t

charges and hence:

= 0

present study, propagates inside optical fiber, both dispersive and nonlinear

effects influence their shape and spectrum. [34, 68]

Which is related to the electric field through the optical susceptibility ,

thro e g r

.8)

Where χ is electric susceptibility, which in linear regime is independent

o whi

From all privious equations , the effect of wavelength and power

refractive index dependancy and also losses due to absorption have been

Where (c ) is light speed in vacuum.

E ctro

(3.4), so we mean consideration wo

o describe the propagation of an optical pulse.

Since fiber is an optical media, therefore, it is free from electric

sources and also it is nonmagnetic, hence:

= = 0

For optical fibers medium, there are no free

For pulse width in range ~10 ns to 10 fs as assumed in the

Due to nonlinear effect (power dependant refractive index), the

induced polarization consists of two parts, linear and nonlinear

contributions as in the following relation [34,55]

( , t)= ( , t) + ( , t) ……………………………………………………………….. (3.7)

ugh th followin elation: [34, 53]

= ε χ ……………………………………………………………. (3

n le in non-linear regime is dependent on [49]


considered . Consequently , second order dispersion , third order

dispersion TOD , nonlinearity and attenuation have been considered and

will appea

oreover the losses in

such laser

……………….. (3. 9)

n medium , g the

average small-signal gain, and, P the average power over one pulse slot

of durati T , which could be calculated as in the following

equation:[13,67]

r later in mode-locked master equation.

Since optical fiber is used as a gain medium (amplifier), the gain

saturation which is another parameters must be considered. Usually the

gain medium in rare-earth-doped fibers, and most solid-state materials,

responds much slower than that of the pulse width. M

s are small and equally distributed. [65]

For this reason, saturation gain g , could be approximated as in

following relation:[21, 48,55,66]

g = g /(1+P /P ) ………………………………

Where, P represents the saturation power of the gai

on

PT |A t, z | dt

T

T……………………………….. (3. 10)

The term A t, z

…

, represents the slowly varying envelope of the electric

d the pulse slot is calculated by following equation:

T = 1/F = TR/N …………………………………………………… (3.11)

s the

ger (N 1 )

field, an

Where F i frequency at which the laser is mode-locked, which is

often denoted F as modulation frequency. N is an inte


representing the harmonic at which the laser will mode locked. TR , is the

rou ip d

The gain medium’s finite bandwidth is assumed to have a

..

T : the sp

Based on previous equations and assumptions , a general"

equation used to model mode-locking fiber laser system is

This equation, is in fact a Generalized Non-Linear Schrödinger

quation GNLSE or (Ginzburg–Landau equation)[1,55] which , generally

just changing the term

M A, t th

A

ndtr an will be identified in next sections.

parabolic filtering effect with a spectral full width at half-maximum

(FWHM) which is given by the following relation: [1, 46]

∆ =2/T . ………………………………………………………… (3.12)

Where:

ectral width of the finite gain bandwidth.

3.2 Mode-locking Fiber Laser Master Equation

master"

introduced.

E

describes all types of mode-locking fiber lasers by

at represents the mode-locker technique. The mode-lock master

equation is: [21,34,47,48]

TR T β ig T LR A – L R

A γLR|A| A g α LRA

M A, t ………………………………… (3.13)

3.2.1 Identifying Mode-locked Master Equation's Terms

The full mathematical model which is able to describes mode

locking for optical pulse propagating through optical fiber , must take in

consideration all parameters that affect on propagated pulse as described in


previous chapter .To verify this fact, let's examine the master Eq. (3.13)

and identify each term, starting from the left side :[45, 52]

z direction,

2nd term describes the effect

The term ig T

1st term describes the basic propagation of the optical field in

of second order dispersion,

3rd term describes the effect of third order dispersion,

4th term describes the effect of nonlinearity

5th term describes the effect of gain and intensity-dependent losses,

6th term represents the mode locker effect which will be identified later.

, results from the gain. The physical origin of this

contribution is related to the finite gain band width of the doping fiber and

is referred

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