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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].
2Chapter One Ultra-Short Pulses Generation
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 pico- seconds. 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]
3Chapter One Ultra-Short Pulses Generation
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)
4Chapter One Ultra-Short Pulses Generation
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.
5Chapter One Ultra-Short Pulses Generation
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]
6Chapter One Ultra-Short Pulses Generation
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]
7Chapter One Ultra-Short Pulses Generation
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]
8Chapter One Ultra-Short Pulses Generation
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.
9Chapter One Ultra-Short Pulses Generation
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]
11Chapter Two Mode-Locking Fiber Lasers
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.
12Chapter Two Mode-Locking Fiber Lasers
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]
13Chapter Two Mode-Locking Fiber Lasers
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
,
14Chapter Two Mode-Locking Fiber Lasers
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.
15Chapter Two Mode-Locking Fiber Lasers
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]
16Chapter Two Mode-Locking Fiber Lasers
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.
17Chapter Two Mode-Locking Fiber Lasers
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.
18Chapter Two Mode-Locking Fiber Lasers
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].
19Chapter Two Mode-Locking Fiber Lasers
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
20Chapter Two Mode-Locking Fiber Lasers
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.
21Chapter Two Mode-Locking Fiber Lasers
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]
22Chapter Two Mode-Locking Fiber Lasers
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]
23Chapter Two Mode-Locking Fiber Lasers
= 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.
24Chapter Two Mode-Locking Fiber Lasers
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
25Chapter Two Mode-Locking Fiber Lasers
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
26Chapter Two Mode-Locking Fiber Lasers
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
27Chapter Two Mode-Locking Fiber Lasers
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.
28Chapter Two Mode-Locking Fiber Lasers
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)
29Chapter Two Mode-Locking Fiber Lasers
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.
30Chapter Two Mode-Locking Fiber Lasers
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.
31Chapter Two Mode-Locking Fiber Lasers
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.
32Chapter Two Mode-Locking Fiber Lasers
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
33Chapter Two Mode-Locking Fiber Lasers
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),
(
34Chapter Two Mode-Locking Fiber Lasers
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.
35Chapter Two Mode-Locking Fiber Lasers
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
36Chapter Two Mode-Locking Fiber Lasers
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.
37Chapter Two Mode-Locking Fiber Lasers
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
38Chapter Two Mode-Locking Fiber Lasers
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
).
39Chapter Two Mode-Locking Fiber Lasers
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
40Chapter Two Mode-Locking Fiber Lasers
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
41Chapter Two Mode-Locking Fiber Lasers
∆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]
42Chapter Two Mode-Locking Fiber Lasers
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.
43Chapter Two Mode-Locking Fiber Lasers
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
44Chapter Two Mode-Locking Fiber Lasers
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
45Chapter Two Mode-Locking Fiber Lasers
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]
46Chapter Three The Model of Mode-Locked Fiber Laser
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 ,
47Chapter Three The Model of Mode-Locked Fiber Laser
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]
48Chapter Three The Model of Mode-Locked Fiber Laser
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
49Chapter Three The Model of Mode-Locked Fiber Laser
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
50Chapter Three The Model of Mode-Locked Fiber Laser
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