carrier heating effects in quantum dot semiconductor ...utq.edu.iq/research/pdf25/30.pdf · we...
TRANSCRIPT
Republic of Iraq
Ministry of Higher Education
And Scientific Research
Thi-Qar University
College of Science
Carrier Heating Effects in Quantum Dot
Semiconductor Optical Amplifier
A Thesis Submitted to the
College of Science Thi-Qar University
In Partial Fulfillment of the Requirements for the Degree of Master
of Science in Physics
By
Salam Thamer Jallod
Supervised by
Dr. Falah H. Al-asady Dr. Ahmed H. Flayyih
(Assistant Professor) (Assistant Professor)
2015 A.D. 1437 A. H
ج
ا
م
ع
ة
ذ
ي
ق
ا
ر
ك
ل
ي
ة
ا
ل
ع
ل
و
م
C
H
3
H
حي حن الره الره بسم الله
إل لن ا علم ل سبحانك ق الوا﴿ العليم أنت إنك علمتن ا ما
﴾الحكيم
صدق اهلل العظيم
23سورة البقرة: اآليت
I dedicate my humble effort
TO
… My father's spirit
… My mother
…My brothers and sisters
Dedication
We certify that we have read the thesis titled "Carrier Heating
Effects in Quantum Dot Semiconductor Optical Amplifier ". Presented by Salam Thamer Jallod, and as an examining committee, we
examined the student on its contents, and in what is related to it, and that in our
opinion it meets the standard of a thesis for the degree of Master of Science in
physics with ( ) degree.
Signature:
Name: Dr. Amin H. AL-Khursan
Title : Professor
(Chairman)
Date : / /
Signature: Signature :
Name : Dr. Hadey K. Mohamad Name : Dr. Shakir D. Al-Saeedi
Title : Assistant Professor Title : Assistant Professor
(Member) (Member)
Date : / / Date : / /
Signature: Signature:
Name: Dr. Falah H. Al-asady Name: Dr. Ahmed H. Flayyih
Title: Assistant Professor Title: Assistant Professor
(Supervisor) (Supervisor)
Date : / / Date : / /
Approved by the Deanery of the College of Science
Signature :
Name: Dr.
Title : Assistant Professor
Dean of College of Science, Thi-Qar University
Examination Committee Certificate
We certify that this thesis entitled "Carrier Heating Effects in
Quantum Dot Semiconductor Optical Amplifier" is prepared by
Salam Thamer Jallod under our supervision at the Physics Department,
College of Science, Thi-Qar University as a partial of the requirements for the
degree Master of Science in physics.
In view of the available recommendations, we forward this thesis for
discussion by the examining committee.
Signature:
Name: Dr. Falah H. Al-asady
Title: Assistant Professor
Date: - -2015
Signature:
Name: Dr. Ahmed H. Flayyih
Title: Assistant Professor
Date: - -2015
Signature:
Name: Dr. Emad Abd uL-Razaq Salman
Title: Assistant Professor
Head of the Physics department
Date: - -2015
Supervisor Certification
I
All praises and thanks are due to Allah, the most beneficent, the most
merciful, for his, graces that enabled me to continue the requirements of this
study.
I would like to express my sincere appreciation to my supervisors,
Dr. Ahmed H. Flayyih and Dr. Falah H. Al-asady, for their continuous help,
valuable remarks, scientific guidance and kindly guidance throughout this
work.
I am grateful to the Head and staff members of the physics department at
the college of science for their support and encouragement, especially
Dr. Amin H. AL-Khursan.
My thanks and great gratitude to all my friends, continuous encouragement
and support that enabled me to overcome many difficulties that faced me
during this research.
I would like to thank my mother, my brothers and sisters. For their support
and patience. Finally, my thanks are also given everyone who helped me in
one way or another,
Salam
Acknowledgements
Abstract
II
Abstract
Theory of carrier heating in quantum dot semiconductor optical
amplifiers did not take enough attention where the most of theoretical models
were processed in classical methods, although the associated phenomenon with
carrier heating are processed in quantum method.
In this study, a new formula has been introduced to study the heating
effect in quantum dot semiconductor optical amplifier for a system composed of
two-level rate equation and depending on density matrix theory and theory of
short pulses in semiconductor material. The investigation of carrier heating
theory has been done through the nonlinear gain coefficient which is considered
the best of techniques to study the nonlinear phenomenon. By depending on the
analytical solution of pulse propagation it has been derived.
The nonlinear gain coefficient due to carrier heating is calculated and
then compared with classical model, it is found that the suggested model agrees
with classical model at value ((1021
-1022
) m-3
) for carrier density, but with the
increasing of carriers above the value (1022
m-3
) the quantum behavior is lower
than the classical model. Also, Carrier heating effect leads to reduce in the
occupation probability, carrier density, nonlinear gain coefficients due to
spectral hole burning, while it is observed that an increase in the time of
recovery with carrier heating occurs. The results of effect of pulse propagation
represented by full width at half maximum that the pulse width is inversely
proportional with occupation probability, carrier density and gain integral, while
it directly proportional with time recovery.
The quantum model of carrier heating reconsiders the theory of four
waves mixing through the interaction between this mechanism and other
nonlinear mechanisms, it obviously shows with the time ( in ) which represents
the effective time of quantum process.
Abstract
III
According to the modified of four-wave mixing, the conversion
efficiency and the symmetric between its components have been studied, the
theoretical results show a good agreement with the experimental data published
in global journals. We indicate that all programs are designed in our laboratory
and they are then, written and solved using a Matlab package.
List of Symbols
IV
List of Symbols
Definition Symbols slowly varying amplitude of the propagating wave A(z, t) absorption renormalized for the occupation probability n ca
the phenomenological parameter to compensate for the nonplanar nature of the waveguide C the velocity of light c total number of states D
the differential gain dg dN electrical intensity of the pump
0E
electrical intensity of the probe 1E
electrical intensity of conjugate formed through nonlinear mixing 2E
electric field of the interacting light E t the input pulse energy
inE the saturation energy
satE the energy difference between the chemical potential E the waveguide-mode distribution function ,F x y
fermi function cf
fermi function at lattice temperature L
cf
material gain g Gain due to spectral hole burning
SHBg
Nonlinear gain due to carrier density pulsation CDP
g Nonlinear gain due to carrier heating
CHg
the maximum value of gain maxg
the small signal gain 0g
Hamiltonian operator of the system H
Hamiltonian for a free atom without perturbation 0H
Hamiltonian at interaction (the interaction of the atom with the field of applied
radiation) H
The gain integral h
Blanck Constants injected current I the current required for transparency
0I
The nonlinear gain coefficient due to carrier heating CH
The nonlinear gain coefficient due to spectral hole burning. SHB
the effective height of the quantum-dot layer the electric dipole moment operator of the atom M
the effective mass for the electrons *
em
carriers density wN
List of Symbols
V
Definition Symbols the grope refractive index
gn
the carrier transparency 0N
the background refractive index bn
the refractive index n
the effective mode index n the instantaneous power of the propagating pulse ,( )P z The power of the input pulse inP
the saturation power of the amplifier sP
the carrier lifetime st
carrier escape time e
carrier capture time c
spontaneous time s
carrier heating time constant CH
decoherent time cv
The full width at half maximum of pump and probe pulses 1 ,
0
the spectral hole time constant. SHB
total intraband time constant in
the pulse energy ( )inU
The energy density U the grope velocity
g
the volume of the active region V angular frequency of the pump
0
angular frequency of the probe 1
angular frequency of the conjugate signal 2
transition frequency i
the linear susceptibility ( )X N Lorentzian lineshape χ( ) decay rate
nm
the optical confinement factor the linewidth enhanced factor
N
the loss coefficient int
the unit vector of polarization the permittivity of free space
0
the dielectric constant the cross section of the active region
m The phase of the input pulse in ,
the phase of the propagating pulse ( ), z
List of Symbols
VI
Definition Symbols the delay time between the two pulses the energy eigen functions
n
the detuning Four wave mixing conversion efficiency
density operator density matrix of motion
nm occupation probability of ground state
c Electron and hole occupation probabilities in the valence and conduction band.
c ,v
dipole term ,cv j wave function
List Of Abbreviations
VII
List of Abbreviations
Meaning Attribute
carrier heating CH
carrier density pulsation CDP
free carrier absorption FCA
four wave mixing FWM
full width at half maximum FWHM
gallium arsenide GaAs
ground state GS
longitudinal optical LO
Linewidth enhancement factor LEF
molecular beam epitaxial MBE
metal-organic chemical-vapour deposition MOCVD
Quantum dot QD
quantum-well QW
Quantum dot semiconductor optical amplifiers QD SOAs
spectral hole burning SHB
Semiconductor Optical Amplifier SOA
two-photon absorption TPA
wavelength conversion WC
Wetting layer WL
cross gain modulation XGM
cross phase modulation XPM
List Of Figures
VIII
List of Figures
Page Description Fig. No. 2 The Semiconductor optical amplifier schematic diagram (1.1)
3 Semiconductor materials used in laser fabrication at different regions of the spectrum (1.2)
4 the SOA structure (1.3)
7 Temporal evolution of carrier distribution after exciting by short optical pulses (1.4)
15 Diagram of carrier relaxation processes in QD (2.1)
16 Semiconductor band structure (2.2)
17 Optical field of pump, probe and conjugate versus frequency (2.3)
40 the GS occupation probability versus carrier density ((4.1
41 the gain versus wavelength (4.2)
42 differential gain and linewidth enhancement factor versus wavelength (4.3)
43 the gain versus carrier density (4.4)
44 the time domain of carrier density (4.5)
44 The occupation probability versus time (4.6)
45 The nonlinear gain coefficient due SHB versus carrier density (4.7)
47 3-dimenssional plot of CH WLN . (4.8)
47 the effect of CH on CH WLN curves (4.9)
48 A comparison between QD model and the Bulk model (4.10)
49 LEF due SHB versus carrier density (4.11.A)
49 LEF due CH versus carrier density (4.11.B)
50 the time domain of gain integral (4.12)
51 the pulse effect on occupation probability (4.13)
51 the pulse effect on the carrier density (4.14)
52 the pulse effect on the gain integral (4.15 )
53 Total FWM efficiency and its components versus detuning (4.16)
53 Total FWM efficiency versus detuning (4.17)
54 Matching between the experimental measurements and our calculation (4.18)
Contents
IX
Contents
Pages Subjects I Acknowledgements
II Abstract
IV List of Symbols
VII List of Abbreviations
VIII List of Figures
IX Contents
1-11 Chapter One: Introduction
1 1.1. Introduction
2 1.2. Semiconductor Optical Amplifiers
2 1.3. Quantum Dot Semiconductor Optical Amplifiers
3 1.4. SOA Material
4 1.5. SOA Structure
5 1.6. Dynamic Recovery in SOA
6 1.7. SOA Nonlinearities
6 1.8. SOA Gain
7 1.9. Literature Review
10 1.10. The Aim of This Work
12-26 Chapter Two: Theory of CH in QD SOA
12 2.1.Introduction
12 2.2.Density Matrix Theory
15 2.3.The Rate Equations
17 2.4.Theory of Nonlinear Process in QD
27-38 Chapter Three: FWM and Pulse propagation
27 3.1.Introduction
28 3.2.Four-Wave Mixing in semiconductor
Contents
X
30 3.3.Theory of Pulse propagation in Bulk SOA
34 3.4.Gain integral and pulse propagation in QD SOA
35 3.5.FWM pulses
37 3.6.The nonlinear Gain Coefficients
38 3.7. Wavelength Conversion in QD
39-54 Chapter Four : The Theoretical Results
39 4.1. Introduction
39 4.2. The Theoretical parameters
40 4.3. Occupation probability of dot level
41 4.4. The gain and differential gain
42 4.5. Transparency Carrier
43 4.6. Dynamic behavior in the time domain
45 4.7. Nonlinear gain coefficients
48 4.8.Linewidth enhancement factor
50 4.9. The gain integral
50 4.10.The dynamic behavior and pulse effect
52 4.11.Wavelength conversion
55-56 Chapter Five : Conclusions and Future Works
55 5.1. Conclusions
56 5.2. Future Works
57-65 References
Introduction
Chapter One
Chapter One Semiconductor Optical Amplifier
1
1.1. Introduction
The first research on semiconductor optical amplifiers (SOAs) has been
begun in the 1960s at the time of the invention of laser. The early devices were
based on gallium arsenide operating at low temperatures [1].
In the late 1960s, the advance of crystal growth techniques, such as
molecular beam epitaxial (MBE) and metal-organic chemical-vapour deposition
(MOCVD), made it possible to fabricate high quality heterostructures with very
thin layers. This has allowed the achievement of low dimensional
semiconductor structures, in addition to studying the impact of quantum size
effect. The use of low dimensional structures has significantly improved the
performance of optoelectronic devices [2].
Quantum dot devices have been predicted to be superior to bulk or
quantum-well (QW) devices in many respects. The fabrication of QD devices
with very low threshold currents [3] indicate effective state filling, which opens
for the potential of making ultrafast QD devices. The two key features necessary
in such devices are high differential gain, which proved to be present in many
QD devices and fast carrier relaxation into the active region.[4,5] which
demorst rated to be about of 100 fs [6].
The dynamic and spectral features of semiconductor lasers and amplifiers
can be calculated by the nonlinear coefficients. Many of studies have been done
to find the origin of gain nonlinearity [7]. Up to now, the physical mechanisms
about the nonlinear gain are still not completely understood, although many of
studies have been devoted to discuss the nonlinear process such as spectral hole
burning (SHB) and carrier heating (CH). It is clear that, through the
experimental and theoretical researches shown the CH and SHB are major
effects on nonlinear gain suppression. The experiment pump-probe, four wave
mixing (FWM) and modulation response reveal the influence of carrier heating
in semiconductor amplifiers [8].
Chapter One Semiconductor Optical Amplifier
2
1.2. Semiconductor Optical Amplifiers
Semiconductor optical amplifiers, as the name suggests, are used to
amplify optical signals. A typical structure of SOA is shown in Figure (1.1).
The basic structure consists of a heterostructure PIN junction. Current injection
into the intrinsic region (also called the active region) can create a large
population of electrons and holes. If the carrier density exceeds the transparency
carrier density then the material can have optical gain and the device can be
used to amplify optical signals via stimulated emission. During operation as an
optical amplifier, light is coupled into the waveguide and then propagates inside
the waveguide it gets amplified. Finally, when light comes out at the end, high
power is obtained [9].
Fig.(1.1): The Semiconductor optical amplifier schematic diagram [9].
1.3. Quantum Dot Semiconductor Optical Amplifiers
Quantum dot semiconductor optical amplifiers (QD SOAs) have great
advantages as compared to bulk and quantum well SOAs, such as high-speed
applications with low-threshold current, high temperature stability and ultrafast
gain recovery dynamics [10]. The QD materials have zero-dimension, and,
theoretically, the energy levels are discrete compare with bulk material [11].
Semiconductor quantum dots have been intensively, theoretically and
experimentally studied in the last years due to their superior characteristice. The
Chapter One Semiconductor Optical Amplifier
3
main feature of QDs is thus the occurrence of discrete energy levels similar to
the ones in atoms. A common quantum-dot semiconductor is not a single layer
device, but several thin quantum-well layers with quantum dots are piled up in
the active region. [12].
1.4. SOA Material
The choice of materials for semiconductor amplifiers depends mainly on
the requirement that the likelihood of radiative recombination should be
adequately high that there is enough gain at low current. This is usually satisfied
for “direct gap” semiconductors. The various semiconductor material systems
along with their range of emission wavelengths are shown in Figure (1.2). Many
of these material systems are ternary (three elements) and quaternary (four-
element) crystalline alloys that can be grown lattice-matched over a binary
substrate. III-V compound semiconductors are composed of group III and group
V atoms, and are mainly used in optoelectronic applications because of their
band structure characteristics. InP, InAs, GaAs, and GaN represent binary alloys
(two elements). Ternary alloys (e.g., InGaAs) are composed of three different
elements. The semiconductors provide the possibility to engineer the band gap
by changing the composition [13].
Fig.(1.2): Semiconductor materials used in laser fabrication at different regions of
the spectrum [13] .
Chapter One Semiconductor Optical Amplifier
4
1.5. SOA Structure
The basic structure of a SOA compose of an intrinsic layer is called the
active, this layer is sandwiched between p-type and n-type material. The carriers
will move from the n-type (electrons) and p-type (holes) towards the active
layer when electrical field is applied over this heterostructure. Here, the carriers
will accumulate as they are trapped in this low band gap potential well. By
applying appropriate pumping , large concentrations of electrons and holes
buildup in the active layer, which leads to population inversion. Photons from
optical pump passing through the amplified medium can stimulate carriers in
the conduction band to relax to their ground state and recombine with holes in
the valence band [14].
The simplest waveguide structures that have been used for the fabrication
of SOAs is a ridge waveguide. It is schematically illustrate in Fig. 1.3. The ridge
waveguide has weakly index guided, and its structure is fabricated to produce a
slight change of the refractive index along the junction plane. Such technique
enhance the optical confinement. In SOAs, the ridge material is usually tilted a
few degrees in order to minimize facet reflectivity. Additionally, a dielectric is
used around the ridge to avoid current diffusion into the p-type layer [15].
Fig.( 1.3): the SOA structure [15].
Chapter One Semiconductor Optical Amplifier
5
1.6. Dynamic Recovery in SOA
After an ultra-short pulse propagate in SOA, a number of dynamic
processes are taken place. They are usually classified into categories; interband
process and intraband process. The recombination of carrier between
conduction and valence band is called interband process. In this process, the
carrier is depletion due to stimulated emission, the recovery time that the
carriers go back to their pervious state is on the order of few hundred of
picoseconds. This process depends on the carrier density, so that is called carrier
density pulsation (CDP).
The dynamics of carrier and intraband transition inside same band can
also occur. The stimulated recombination burn a hole in carrier distribution
making it deviate from the fermi distribution, this process is called spectral hole
burning (SHB), the time recovery (carrier-carrier scattering time) is the carrier
go back to the fermi distribution is about tens of femtoseconds.
Carrier-carrier scattering also take place due to two-photon absorption
(TPA), where the strong sub-picosecond pulse will be created free carriers by
consumption of two photons. These generated carriers are of a higher
temperature than the lattice and cool-down in a sub-picosecond time scale
( . Additionally, Free-carrier absorption is an optical absorption
process that does not generate electron-hole pairs; instead, the photon energy is
absorbed by free-carriers in either the conduction or valence band, moving the
carrier to a higher energy state within that band. The carrier temperature is
increased in this process and cools-down to the lattice temperature in a time
scale of hundred of femtoseconds. This process is denoted carrier Heating (CH)
[15].
Chapter One Semiconductor Optical Amplifier
6
1.7. SOA Nonlinearities
The propagation of pulse in SOA is followed by a number of nonlinear
interactions that make the semiconductor have high nonlinearity. They grow to
be especially strong by the use of short optical pulses and/or by exerting strong
optical powers. The nonlinear behavior in semiconductor comes from the
dependence of the susceptibility to the applied optical field [1]. Nonlinearities in
SOAs are principally caused by carrier density changes induced by the amplifier
input signals. Four-wave mixing and cross gain modulation (XGM) are the most
nonlinear processes which can be exploited for wavelength conversion (WC)
[17,18].
1.8. SOA Gain
The dynamical processes that determine the gain variation after
propagating an optical pulse through the SOA can be classified into interband
and intraband process, the gain coefficient may be expressed as [18]
(1.1)CDP SHB CH
Interband Intraband
g g g g
where g is total gain, CDPg is the gain due to (CDP) which is an interband
processes (e.g. spontaneous emission, stimulated emissions and absorption)
depending on the carrier density, while CHg and SHBg are the intraband
contributions of gain (CH and SHB). The interband process refers to the
recombine of carriers between the conduction and valence bands which affects
the carrier density and the energy gap, which is a slow process with a time
constant in the range of part of nanosecond. Interband mechanism dominates the
SOA dynamics when long optical pulses are used. On the other hand, when the
SOA is operated using pulses shorter than few picoseconds, intraband effects
become important. The carriers distribution change within same band. The short
pulses cause reduction of carrier distribution as in Fig. (1.4). (a deviation from
Chapter One Semiconductor Optical Amplifier
7
the Fermi distribution), the time needed to restore the Fermi distribution by
scattering processes (mainly carrier-carrier scattering) is called SHB time
constant. The increasing of carrier temperature above the lattice temperature
will change carrier distribution, after several hundreds of femtoseconds to few
picoseconds the carriers will restore its distribution and cools down to the lattice
temperature through phonon emission.
Fig.(1.4) Temporal evolution of carrier distribution after exciting by short optical pulses
[20].
1.9. Literature Review
This section provides an overview of pervious works related to this study.
Since then, several approaches to achieve CH in SOA have been experimentally
and theoretically produced, some of the interesting works related with the topic
of thesis are:
M. Willatzen et al ([21], 1992): the present numerical results for nonlinear gain
coefficients due CH and SHB for QW laser. The small signal analysis is achieve
for temporal evaluation of carrier and photon densities to obtain information
about nonlinear gain coefficient. This study was concluded, that CH is an
Carrier Density
En
erg
y
Electrical Pumping
SHB Heating Cooling
Chapter One Semiconductor Optical Amplifier
8
important mechanism for nonlinear gain coefficient and its effect increased
faster with strain.
F. Jahnke and S. W. Koch ([22], 1993): in this study, the nonequilibrium
carrier distributions in laser was determined by solution of a quantum
Boltzmann equation including carrier–carrier, carrier–phonon, and carrier–
photon scattering as well as the pump process. A significant heating of the
carrier plasma is observed as a consequence of the Pauli blocking of carrier
injection and the removal of cold carriers through the process of stimulated
recombination.
A. Uskov et al ([23], 1994): they modeling four-wave mixing in QW SOA,
including the effect of carrier density pulsation, SHB and CH, The equations
derived based on density matrix theory and then solved numerically. The
theoretical results explain different experimental ones, which have been taken
into account the role of CDP, SHB and CH in FWM.
C. Tasi et al ([24]. 1995): In this study, the hot phonon effect on the CH and
the saturation of resonant frequency in high speed QW laser are investigated
theoretically.
A. Uskov et al ([25], 1997): They introduced a numerical model to studying the
carrier cooling and carrier heating in bulk semiconductor, the saturation
dynamic and pulse propagation were also investigated, the saturation causes
reduction in the saturation energies for sub-picosecond pulses in comparison
with picosecond pulses. Comparison of bulk and QW absorbers shown that fast
saturation could be stronger in a bulk absorber, so bulk saturable absorbers may
be interesting for usage in mode-locked solid-state laser.
J. Wang and H. Schweizer([26], 1997): In this study, a comparison of the
classical rate-equation model with the carrier heating model have been done for
quantum-well (QW) lasers, the contributions of the dynamic of carrier and
energy relaxation in nonlinear phenomenon are investigated. This study shown
the contribution of CH to the nonlinear gain coefficient is proportional to an
Chapter One Semiconductor Optical Amplifier
9
effective carrier energy relaxation time, and the contribution of the electron-hole
energy exchange time show a nonlinear behave, Furthermore, the Auger heating
effect on the modulation dynamics is also considered.
T. Sarkisyan et al ([27], 1998): This paper used modified rate equations to
described the macroscopic behavior of a semiconductor laser. This model takes
into account the nonlinear functional dependence of the gain coefficient on
carrier density and temperature.
X. Yang ([28], 2003): The propagation of short pulses in SOA have been
modeling using rate equations, the theoretical results is shown a good
agreement with experiment results.
Y. Ben-Ezra et al ([29], 2005): they introduced a new technique for reducing
the patterning effect in QD SOA by using an additional light beam. The
theoretical analysis of the carrier dynamics in QD-SOA is presented. It is shown
that the increase of the current only partially improves the QD-SOA temporal
behavior. The additional light beam drastically reduces the patterning effect.
O.Qasaimeh ([30], 2008): the researcher introduced a closed-form model for
multiple-state QDs SOA taking in to account the effect of ground state (GS),
exited state (ES) and WL. The analytic solution was shown that the effective
saturation density of QD-SOAs strongly depends on the photon density and the
biased current.
D. Nielsen et al ([18], 2010): This study introduced an analytical model to
determine FWM in QDs base on density-matrix formalization for single bound-
state QDs, the theoretical results was shown a good agreement with experiment
and revealed that there is a significant contribution from carrier heating in the
FWM efficiency.
N.Majer ([31], 2011): The impact of carrier-carrier scattering on the gain
recovery dynamics of a QD SOA have been investigated. Coulomb scattering
rates between WL and QDs have been calculated by using Bloch equations. The
simulation shown a good agreement with experimental measurements.
Chapter One Semiconductor Optical Amplifier
10
A.H. Al-Khursan et al ([32], 2013): The theory of FWM in the QD-SOAs is
discussed by combining the QD rate equations system, the quantum-mechanical
density-matrix theory, and the pulse propagation in QD SOAs including the
three region of QD structure GS, ES and wetting layer. It is found that inclusion
of ES in the formulas and in the calculations is essential since it works as a
carrier reservoir for GS. It is found that QD SOA with enough capture time
from ES to GS will reduce the SHB component, and so it is suitable for
telecommunication applications that require symmetric conversion and
independent detuning.
H. Al-Khursan et al ([33], 2013): In this paper, a new formula of integral gain
in QD-SOAs depending on the QD states has been derived instead of
conventional bulk relation. Wetting layer, ES and GS of SOAs have been
employed to study the effects of important parameters in such these devices.
Good results were obtained, since the effective capture time in QD is controlled.
A. H. Flayyih and A.H. Al-Khursan ([34], 2014): the effect of CH in the
FWM theory in QD structure has been studied. The influence of parameters
such as CH nonlinear gain parameter, wetting layer carrier density, CH time
constant, QD ground and excited state energies have been examined. The model
predicts a low CH for QDs which can explain earlier experimental
measurements in this field.
1.10. The aim of this work
Carrier heating phenomenon in semiconductor materials has been known
in the late of 1980 and many researchs are reported to studying its effects, where
the nonlinear gain coefficient due CH is used to describe heating effects in SOA
[18]. The theoretical models to studying CH in QD is not taking much interest
and the equations derived in QW for simulate CH were used for modeling of
CH in QDs (for example, [18, 20]). So the aim of this thesis is introduced a new
Chapter One Semiconductor Optical Amplifier
11
formula to simulate CH in QD taking into account the feature of QD structure.
To satisfy this objective, we must do the following steps:
1. Deriving the equation of polarization based on density matrix equations.
2. Extract the susceptibility of carrier heating effect from the other
components of polarization (CDP and SHB).
3. Deriving the nonlinear gain coefficient due to carrier heating by
normalizing the nonlinear susceptibility of carrier heating depending on
the analytical solution of pulse propagation inside QD SOA.
Theory of Carrier Heating in
Quantum Dot
Semiconductor Optical Amplifier
Chapter Two
Chapter Two Theory of CH in QD SOA
12
2.1. Introduction
The effects of carrier heating (CH) in semiconductor optical amplifier (SOA)
is not less important than spectral hole brining, it is an intraband process and
affect strongly the gain dynamic of bulk and quantum well (QW) with sub
picosecond time scale [35-36,26], make a strong contribution to the high-speed
performance of the devices [14,37] . The main sources of CH are injected
carrier from barrier to dot structure [38], carrier recombination, where the “cold
carriers”, which are close to the band edge, are removed [23], and free carrier
absorption, which includes the photon absorption by the interaction of free
carrier within the same band [16]. In all of these processes the temperature of
carriers will be higher than the lattice temperature. To reach thermal
equilibrium, the carriers will transfer the access energy to the lattice through the
interaction with phonons [39].
Carrier heating effect has been studied by nonlinear gain coefficients, which
affects strongly the maximum modulation bandwidth and wavelength
conversion [18,32,40-41]. Carrier heating influence on the performance of
lasers and amplifiers in bulk and QW has been reported by a number of work
[31,32,42], and used the CH nonlinear gain coefficient in QW to modeling the
conversion efficiency of four-wave mixing in QD [18]. The theoretical study of
Auger capture induced by CH in QD has been introduce by Uskov et .al [37].
This section presents a new theoretical model to simulate CH in QD
structure depending on the density matrix theory and rate equations of two-level
rate equation. Our development is not just about the theory of CH in QD SOA,
but it includes other nonlinear processes such as (CDP and SHB).
2.2. Density Matrix Theory
The density-matrix theory plays an important role in applications to linear
and nonlinear optical properties of materials in quantum electronics. The basic
idea is that the density-matrix formulation provides the most convenient method
Chapter Two Theory of CH in QD SOA
13
to predict the expectation values of physical quantities when the exact wave
function is unknown. The mathematical expression of density operator is given
by the following equation [43]
(2.1)
Where is the wave function which obeys the Schrödinger equation [43]
(2.2) H it
ħ
where H denotes the Hamiltonian operator of the system. We assume that H
can be represented as [43]
0 (2.3) H H H
where 0H is the Hamiltonian for a free atom without perturbation and H is
represent the Hamiltonian of the interaction (the interaction of the atom with the
field of applied radiation), This interaction is assumed to be weak in the sense
that the expectation value and matrix elements of H are much smaller than that
of 0H . It is usually assumed that this interaction energy is given as [43]
. 2.4 H M E t
where ( )M er denotes the electric dipole moment operator of the atom, E t is
the electric field, e is the charge of electron, r is the distance between the charge .
Assume that the states n represent the energy eigen functions n of the unperturbed
Hamiltonian 0H and thus satisfy the equation 0 n nnH E . As a consequence, the
matrix representation of 0H is diagonal that is [44],
0, 2.5nm n nmH E
The commutator can be expanded and the summation over ν can be performed
formally to write as[43].
0 0 0 0, 0,, ( ) ( )
( )
2.6
nm nv vm nv vmnmv
n nv vm nv vm m
v
n m nm
H H H H H
E E
E E
Chapter Two Theory of CH in QD SOA
14
The transition frequency (in angular frequency units) is
2.7n m
n m
E E
The density matrix equation of motion with the phenomenological inclusion of
damping is given by [43]
nm, 2.8eq
nm nmmnm n
iH
Substituting Eqs. (2.3, 2.6 and 2.7) in Eq.(2.8), the equation of motion is
nm nm, 2.9eq
nm nm nmnm mn
ii H
The expanded of as a linear combination [44]
(0) 1 (1) 2 (2) ... (2.10)nm nm nm nm
The solutions of Eq.(2.9) are [44]
(1)
(0) (0) (0) ( )
(1)
(
1 (0)
nm
(2) 1 (0)
n
2
m
)
2.10A
, 2.10B
,
nm nm
nm
eq
nm nm nm nm
nm
nm
nm
nm nm
i
i i H
i i H
2.10C
Equation (2.10A) describes the time evolution of the system in the absence of
any external field. We take the steady-state solution to this equation to
(0) ( )eq
nm nm [44] where ( ) 0eq
nm , Now that (0)
nm is known, Eq. (2.10B) can be
integrated. To do so, we make a change of variables by representing (1)
nm as [44]
(1) (1) ( )e 2.11nm nm
nm nm
i tt S t
The derivative of (1)
nm can be represented in terms of (1)
nmS as [44]
(1) (1) (1)( )e ( )e 2.12nm nm nm nmi t
nm nm
i t
m nmnm ni S t S t
These forms are substituted into Eq. (2.10B), which then becomes [44]
Chapter Two Theory of CH in QD SOA
15
(1) (0)
nm , e 2.13nm nm
n
i
m
tiS H
This equation can be integrated to give [44]
(0)(1)
nm( ) , e dt 2.14nm nm
t
nm
i tiS H t
This expression is now substituted back into Eq. (2.11) to obtain [44]
( )(0)(1
n
)
m( ) , e dt 2.15nm nm
ti t t
nm
it H t
2.3. The Rate Equations
The model represents carrier dynamics in two-level system (WL, GS) can be
seen in Fig.(2.1)
Fig. (2.1) Diagram of carrier relaxation processes in QD [18].
The rate equations describe dynamic of carriers for QD 2-level system
including CH contribution is given as [18]
(1 )(2.16)
(1 )12 ( ) ( ) (2.17)
w c w c w
e c s
c c w c c cc
e c s CH
dN N N ID
dt eV
d Na E t
dt D
c is the occupation probability of ground state, Nw is the carriers density, e
and c are the carrier escape and capture times, respectively, D is the total
number of states, s is the spontaneous time, CH is the CH lifetime , I is the
GS (QD Level)
τc τe
CB
VB
Chapter Two Theory of CH in QD SOA
16
injected current, E(t) is the electric field of the interacting light, n ca is the
absorption renormalized for the occupation probability, according density
matrix theory na is given as [18]
, , , ,( ) (2.18)2
c vc i cv i cv i vc i
i
ia
V
According density matrix theory, the transition between conduction band (CB)
and valence band (VB) is govern by the following equation [18]:
1( ) ( 1) ( , ) (2.19)
cv
cvcv c v
cv
d ii E z t
dt
cv is decoherent time, i is transition frequency, the transition energy is given
by [23]
,, (2.20)vd ii cd i gE E E
The band gap-shrinkage effects (the dependence of gE on carrier density)
[45] is neglected, which is a good approximation under typical laser operating
conditions [46]. These equations essentially treat the semiconductor as being
composed of an inhomogeneously broadened set of two-band systems, c and
v denote the electron and hole occupation probabilities in the conduction and
valence bands, respectively.
Fig. 2.2: Semiconductor band structure [23].
Chapter Two Theory of CH in QD SOA
17
2.4. Theory of Nonlinear Process in QD
To study Nonlinear process in SOA, we are using the wave-mixing model
to study effects of SHB and CH, the wave mixing is achieved by exposing the
SOA to the strong optical field (pump) at an angular frequency 0 and a weaker
probe at 1 , the fields can be mixed nonlinearly to produce a so-called
conjugate signal at 02 1( )2 as in Fig 2.3. consider a total electric field
propagating in the SOA of the form [23]
1 2
0 1 2( , ) ( ) ( ) ( ) . (2.21)oi t i t i tE z t E z e E z e E z e c c
0E is the electrical intensity of the pump, 1E is that of the probe and 2E is the
conjugate formed through nonlinear mixing [23].
The field ,E z t induces a polarization ,P z t in the active region of the
amplifier [23]
1 2
0 1 2( , ) ( ) ( ) ( ) . (2.22)oi t i t i tP z t P z e P z e P z e c c
The relation that relate between the polarization and dipole terms is given by
[18]
,
,
1, 2.23cv j cv vc
j i k
P z tV
The dipole terms take the form [23]
1 2
, ,0 ,1 ,2 (2.24)oi t i t i t
cv j j j je e e
0 2 1
Optical Field
Fig. 2.3: Optical field of pump, probe and conjugate versus frequency [23].
Chapter Two Theory of CH in QD SOA
18
As the pump light field is assumed to be on resonance with the QDs
however, the contributions from continuum’s k states can be ignored. The
small-signal analysis for carrier density and occupation probability is [47,49]
*
, , , , (2.25)i t i t
c j c j c j c je e
*
, , , , (2.26)i t i t
w j w j w j w jN N N e N e
The variables on the right hand side of the Eqs. (2.25) and (2.26) are time
independent, the solution of Eq. (2.24) is achieve by substituting Eqs. (2.25 and
2.26) in Eq. (2.19), one obtain [23]
0
1
2
, * *
, , , , , 2 , , 1
,
1 , , 1 , , 0
, * *
2 , , 2 , , 0
( ) ( 1) ( ) ( )
( ) ( 1) ( )
( ) ( 1) ( ) (2.27)
cv j i t
cv j j o c j v j o c j v j c j v j
cv j i t
i c j v j c j v j
cv j i t
i c j vd i c j v j
E E E e
E E e
E E e
where ( )j is the Lorentzian lineshape determined by the decoherence time
and is responsible for homogeneous broadening [23]
1( ) 2.28
( )j
j i
The dipole terms, Eq. (2.24), can be extracted by the comparison between Eqs.
(2.27) and (2.24), to get [23]
, , , , 2,
,0 * *
, , 1
( 1) ( )( ) 2.29
( )
c j v j o c j v jcv j
cv j o
c j v j
E E
E
,
,1 1 , , 1 , , 0( ) ( 1) ( ) (2.30)cv j
cv j c j v j c j v jE E
, * *
,2 2 , , 2 , , 0( ) ( 1) ( ) (2.31)cv j
cv j c j v j c j v jE E
( , ,( 1)c j v j and , ,( )c j v j ) are considered as Fermi function at
steady state and small-signal analysis, For the steady-state solution, the
combination of Eq. (2.17) for conduction and valance band, one obtains
Chapter Two Theory of CH in QD SOA
19
, ,2 2*
0 0 02
21
1 (2.32)2
1 ( ) ( )
win
c
c j v jin
cv j j
N
D
iE
Where
1
1 1 1(2.33)wv
in
c e CH
N
D
When the pump is turned off it is expected that the dot occupation
probabilities should be the same as the occupation probability under thermal
equilibrium, F, such that , , 1 1c j v j c vF F [18]. By taking 0 0E in
Eq. (2.32), one obtains
2
1 1 2.34wc v in
c
NF F
D
The derivation of eq. (2.34) gives
,, 2 11 2.35
v jc j in inw
w w c c
ddN
dN dN D D
By comparing the result in Eq. (2.33) with the result in reference [18] for
QD SOA, here, the above equation is differ from ( Eq. (28) in [18]) by the term
of carrier heating time constant (CH ). In QD structure, the time of spectral-hole
burning time constant is equivalent
1
1wv
c e
N
D
which represents the rate at
which the quantum-dot ensemble will relax to thermal equilibrium via these
capture and escape dynamics and it limits by carrier densities in wetting layer,
so that in can be considered as a total intraband time constant.
The calculation of carrier heating process with synchronization of
spectral hole burning effect is require taking in to account Fermi function
relaxation in our calculation and according to the density matrix theory
Eq.(2.17) can be written as [18, 23]
Chapter Two Theory of CH in QD SOA
20
(1 )12 ( ) ( )
(2.36)
L
c cc c w c c c ccc
e c s CH
ff N f fda E t
dt D
Where fc is the Fermi function which is a function of temperature
( ) (2.37)cf t F T t
is Fermi function at lattice temperature, the steady state and small signal of
Fermi function can be written as [23]
,
, ( exp( )) (2.38)c k
Fc c k c
c
ff f T i t
T
The small signal of occupation probability accompanied by the thermal
relaxation in QD SOA is not derived earlier [for example; see ref. [18] ].To
derive this probability with existing carrier heating effect, substituting
Eqs.(2.25, 2.26 and 2.38) in Eq. (2.36) and combine for conduction and valence
bands, one obtain
, , , ,
, v,
, ,
22
* * * *
, , 1 0 1 0 0 2 0 22
1 1 1 1 1 1 1( ) ( )
1 1 1( )
1 χ ω χ E χ ω χ E )ˆ ˆ ˆ ˆ
w w wc k v k c k v k
e c CH c e c CH
c k kw wc k v k c v
c e c CH c v
cvk
c k v k
N N Ni
D D D
f fN Nf f T T
D D T T
iiE E
c,k v,k2
2*
1 2 0ˆ ˆ (2.39χ )
ρ ρ
χ ω E
ħ
, v,
, , , ,
2
* * * *
, , 1 0 1 0 0 2 0 22
1 1 1{ [( ) 1 ( )] ( )
(1 )
21 χ ω χ E χ ω χ Eˆ ˆ ˆ ˆ )
(2.40)
c k kw wc k v k w c k v k c v
in c c c v
in k
c k v k
f fN NN T T
i D D T T
iE E
Chapter Two Theory of CH in QD SOA
21
The small signal of Eq. (2.16 and 2.40) and using Eqs.(2.25, 2.26 and 2.38), the
result
2
* * * *
1 0 1 0 0 2 0 22
2
2 *
0 1 22
2
2 *
0 1 22
2
2
0 12
211 χ ω χ E χ ω χ E )*
2{1 E χ ω χ }
221[(WY XZ) E χ ω χ [1 ]
1
ˆ ˆ ˆ ˆ
ˆ ˆ
ˆ
E χ ω
ˆ
ˆ χ
i w inw
i c
in k
i i
i w inini i
i c c
i
i
i
NiN X E E
D V D
i
NiX
D V D D
iX
D V
, v,*
2
2
2 *
0 1 22
( )
221[(WY XZ) E χ ω χ [1ˆ ˆ ]
(2.41)
c k k
i c v
c v
i w inini i
i c c
f fT T
T T
NiX
D V D D
Where
1 1 1 1(2.42)w
e c s CH
NY i
D
11(2.43)c
c
ZD
1 1 1( ) (2.44)c
e s c CH
W i
The linear and nonlinear polarization (P0, P1 and P2) is simply estimated
by substitute of the dipole terms in Eq. (2.23), then, comparing it with Eq.
(2.22). Separating terms of different resonances, the orders of polarization are
given by [23,18]
2
,
0 , , 0
1( )( 1) 2.45
cv j
j o c j v j
j
P EV
2
,
1 1 , , 1 , , 0
1( ) ( 1) ( ) (2.46)
cv j
j c j v j c j v j
j
P E EV
2
, * *
2 2 , , 2 , , 0
1( ) ( 1) ( ) (2.47)
cv j
j c j v j c j v j
j
P E EV
Chapter Two Theory of CH in QD SOA
22
Taking these expressions and combining them with the earlier expressions for
the polarization densities, it found that the pump polarization density and linear
susceptibility, , to be by using equation (2.32) written as
2
0 0 02, 2 *
0 0 02
21
1P ( ) χ ω ( ) (2.48)
2{
ˆ
ˆ1 E χ ω χ }ˆ
w in
j c
j i k
j
in k
i i
N
Dt E
V i
The linear susceptibility is simply extracted by comparison with 0 0 1P E
2
( )
0 2, 20 *
0 0 02
21
1 1χ ω ( ) (2.49)
2{
ˆ
ˆ ˆ1 E χ ω χ }
w in
i cl
j i k in
j
k
i i
N
DX
V i
Substituting Eqs.(2.40) and (2.41) in Eq.(2.46), the second-order polarization is
given by
2
(l)
1 0 1 1 1 , ,
,
2
, v,
1
,
2 2
1 , ,2,
*
1 0
1 1 1P ( ) ω E χ ω { [( ) 1
(1 )
1 1 1( )]} χ ω {( )}
(1 )
21 1χ ω 1
(
ˆ
ˆ
ˆ1
ˆ
)
χ χ( ω
i inw c k v k
j i k in c
c k kiinc v
j i kc in c v
i in k
c k v k
j i k
j
in
j
j
t NV i D
f fT T
D V i T T
i
V i
* * *
1 0 0 2 0 2 0E χ ω χ E )E (2.50ˆ ˆ )E E
The induced polarization density in Eq.(2.50) contains four terms, the first term
is represent the linear polarization density associated with gain or absorption in
the optical amplifier, the second is the nonlinear interaction between the pump
and probe due to CDP, the third term is the nonlinear interaction due to CH and
the last term is the nonlinear interaction due to SHB. The polarization density
2P is identical to that of 1P . For simplifying the density polarization is expressed
as [18,23,34]
Chapter Two Theory of CH in QD SOA
23
2*0
1 0 1 1 0 1 0 1 1 0 1 2 0 22
0
2*0
0 1 0 1 1 0 1 2 0 22
0
2*0
0 1 0 1 1 0 1 2 0 22
0
; ; ; ;
; ; ; ;
; ; ; ; (2.51)
CDP CDP
L
SHB SHB
CH CH
EP X E X E X E
E
EX E X E
E
EX E X E
E
2*0
2 0 2 2 0 2 0 2 2 0 2 1 0 12
0
2*0
0 2 0 2 2 0 2 1 0 12
0
2*0
0 2 0 2 2 0 2 1 0 12
0
; ; ; ;
; ; ; ;
; ; ; ; 2.52
CDP CDP
L
SHB SHB
CH CH
EP X E X E X E
E
EX E X E
E
EX E X E
E
The various contributing terms are separated from each other. These
factors include the linear and the nonlinear effects. The susceptibility due CDP
and SHB can be simplified using the definition of gain and differential gain
which are expressed by
2
*
0
1 21 ( ) ( ) 2.53
2
j win j j
j c
Nig
c n V D
2
0
1 2 1( ) 1 ( ) 2.54
j in inw j
j c c
dgi N
dN c n V D D
Eqs.(2.53) and (2.54) have been derived by substituting the relation of
, , 1c j v j and , ,c j v j
w w
d d
dN dN
in QD system identified into Eqs.
(2.34,2.35). These identities also introduce an important parameter which can be
compared to experiment including linewidth enhancement factor, N , the
refractive index, n , and the material gain, g , which is calculated from the
susceptibility defined in Eq (2.49). Substitute Eqs. (2.41, 2.53-2.54) into Eq.
(2.50) and then compare the result with Eqs.(2.51-2.52), we can determine
generalized susceptibilities due to CDP and SHB as
Chapter Two Theory of CH in QD SOA
24
2
2
0 0 0
0
2
0
22
( )2( )
; ;1
(2.55)
q
sqqCDP
q m n
nm ins
l
sat
dgcng E
idNX
i D EWY XZ
X E
2
0
0
4
*
3
2; ;
1
2( ) 1 ( ) ( ) (2.56)
SHB inq n m
mn in
wj q in j m j n
c
EiX
V i
N
D
2l
satE is the saturated field for two-level QDs system which can be defined as [18]
2
20
2.57
2
l
satl
s
w
Edg
c ndN
The susceptibility due CH is derive as
2
, v,
,
1 1χ ω { }
(1 )ˆ( , , )
(2.58)
c k ki
q c v
j i k i
CH
q m n j
n mn c v
f fT T
V i T TX
To calculate the temperature at small signal, we must use the definition of
energy density (U) which is given by the following equation [23]
, , (2.59)x x j x j
j
U t t
Multiplying (2.2) by ,x j , and summing over j, one obtain
2xvc,i cdvd,i cv,i cdvd,i
( E ) 1( , ) ( ) E(t)
(2.60)
wx x
ie c s CH
N UU U U iU K E z t E
D V
The term (2
( , )xK E z t ) is phenomenologically added to represent the
contribution of CH induced by FCA, is a coefficient that can be express by
Chapter Two Theory of CH in QD SOA
25
the cross section for free carrier absorption (FCA) in the conduction and
valence bands which is given by [18]
0 (2.61)x g g x wK n n N
Where x is the cross-section,
0 is the permittivity of free space, n is the
refractive index, gn is the group refractive index and g is the group velocity. x
refers to conduction (c) or valance band (v). To determine the expression of
temperature at small-signal, we use the expansions [23].
. (2.62)i t
x x xU U h T e c c
Substituting Eq. (2.62) and Eq.(2.21)in Eq.(2.60) , to obtain
2
1 0 0 2 2
* * * *
1 0 1 0 0 2 0 2
2
, ,
2
exp 1 exp exp
exp exp 12 ( )
ˆ ˆ ˆ ˆ(
exp
.{ χ ω χ E χ ω χ E }exp1)
(1
x x wx x x x
e c
x x x x cv
x
s CH
c k vx
cv
x
k
h T i t Ni h T i t h T i t
D
h T i t h T i t iK E E E E i t
V
E E E i t
iE
V
2*
1 2, 0, χ ω χ exp
(2.
ˆ ˆ)
63)
c k v k E i t
For simplicity, we neglect the last term in Eq.(2.63) and used Eq.(2.18), then
one gets
2
1 0 0 2 2
* * * *
1 0 1 0 0 2 0
, ,
2
11 2 ( ) .
{ χ ω χ E χ ω χ E } (2.64
( 1)
ˆ ˆ ˆ )ˆ
in cv
x x in in x c k v kx
iT h i K E E E E E
V
E E
21* * *
0 0 1 0 0 22
1 0 0 2
, ,
1{ χ ω χ (E E )
1
2 ( )} (2.6
( )
5)
ˆ ˆ1cvin x
x x
in
c k v k
x
ihT E E E
i V
K E E E E
Chapter Two Theory of CH in QD SOA
26
From definition of the material gain (Eq.(2.53)) and free carrier absorption
factor, Eq.(2.61) can simplified as
1
00 0 g 1 0 0 2
0
2{ g( ) 2 }( ) (2.66)
1
in xx x g X
in
chT E N E E E E
i
Substituting Eq.(2.66) in Eq.(2.58) and use the identity
2
,
0 0 0
ω ω1χ ω ( ) ( (ω) ) (2.67ˆ )
x
x kcv
T
k x x x
f gci
V T T T
The susceptibility due to CH will become
10
0 g 0
1 0 0 2
0
( (ω) ) 2ωg( )
1 1
[ ]( ) (2.68)g( )
xT gCH in x
x in in
X wx
igc h
T i i
NE E E E E
Four Wave Mixing
and
Pulse propagation
Chapter Three
Chapter Three FWM and Pulse Propagation
27
3.1. Introduction
Nowadays, in high-speed communication systems, all-optical signal
processing techniques play an important role to avoid electro-optic conversion
which create data flow bottleneck [50]. One of these ways which can be used is
four-wave mixing (FWM). It is a promising technique that can replace multi-
wave converters by a single one [51]. It is typically realized in semiconductor
optical amplifiers (SOAs) and requiring an external pumping sources [52].
SOAs contain low-dimensional structures such as quantum dot (QD) in their
active region gets considerable attention due to the possibilities offered by QDs.
This includes: excellent controllability of intraband transitions which have been
essential in optical devices, ultrafast response unlimited by carrier
recombination lifetime [53], in addition to the promising properties such as low
threshold current, temperature insensitivity, high bandwidth, and low chirp [54].
All of these characteristics make QD SOAs a promising candidates for devices
used in fast and all-optical manipulations [55].
FWM results from nonlinear interaction between two waves differ in
frequency and intensity inside a semiconductor. The beating of two waves
results in a new waves as a result of modulation of both gain and refractive
index and a generation of diffraction grating [56]. The mechanisms that lead to
FWM in semiconductors are includes carrier density pulsation (CDP), carrier
heating (CH) and spectral hole burning (SHB) [57]-[59]. Since QD SOA active
region has a totally quantized QDs grown on a two-dimensional wetting layer
(WL), there are a differences appear in FWM processes in QD SOAs from that
of bulk SOAs [60]. The SHB is governed by the carrier-carrier scattering rate
where the optical field digs a hole in the intraband carrier distribution due to
stimulated emission [57]. Here in QD SOAs, to return to quasi-equilibrium in
QDs, intersubband and interdot relaxations must occur. The relaxation from WL
to QD is slow as a result of transition from two-dimensional WL to completely
quantized QD states [18]. It is on the order of picoseconds [18], [61]. This is the
Chapter Three FWM and Pulse Propagation
28
well-known phonon bottleneck effect [62]-[64]. CDP is governed by the
radiative recombination time which is on the order of nanosecond. It results due
to beating between the pump and signal, then, carriers depletes near the signal
wavelength thus, reducing the overall gain spectrum [65]. CH is governed by
two characteristic times: carrier-carrier and carrier-phonon scattering times.
While free carriers increases their energy states by absorb photons, the
stimulated emission removes lowest energy carriers thus, raising the carrier
temperature. The hot carriers cools down by carrier-phonon collisions [18],[65].
Since QDs shows a reduced carrier density due to discrete energy subbands,
thus it is demonstrated experimentally that QDs have a reduced carrier heating
compared with bulk and quantum-wells [66]. FWM in QD SOA has been
studied [18,20,30,32,67] , in theoretical models use various approximations . In
this thesis we develop a general theory of FWM, also the effect of the pulse on
the FWM efficiency in QD SOAs is not takes an enough length in researches.
Thus, a detailed study combines the theory of pulse impinging on the QD SOA
and its effects on the FWM results from impinging and SOA wave is required.
This work deals with a new model to simulate FWM in QD SOA, also the
influence of the pulse propagation in QD SOAs has been included in this study.
3.2. Four-Wave Mixing in semiconductor
In the optical regime the field interacts with the medium in a number of
linear and nonlinear ways. In linear process the polarization induced by the
field is proportional with the first order susceptibility, where the waves are pass
through each other in the medium without influencing each other and no
coupling of wave occur. Nonlinearity arises when the polarization becomes
proportional to the higher order of field, these effects have been observed after
the invention of laser because they are observable only with high intensities
[68]. Wave mixing arises from the nonlinear optical response of a medium
when more than one wave is present. It results in the generation of another wave
Chapter Three FWM and Pulse Propagation
29
whose intensity is proportional to the product of the interacting wave intensities
[69].
FWM has three different physical mechanisms contributing toward its
conversion. The first mechanism is called carrier density pulsation, the beating
between the pump and probe allowing wave mixing by producing a temporal
grating in the device. The CDP is interband process like cross-gain modulation
and cross-phase modulation and is thus limited by the recombination and
generation rates of carriers [18]. However, four-wave mixing also has
contributions due to spectral-hole burning and carrier heating, which are
governed by the scattering process such as carrier-carrier and carrier phonon-
scattering, the fast rates of this process can be exploiting in higher speed
devices. With strong pump, spectral hole burning is occur where the carriers are
depleted. After a time of about tens of femtoseconds the carriers are relax down
into the depleted states via carrier-carrier scattering and the system will be in
quasi-equilibrium.
The last FWM mechanism is carrier heating, in which the temperature of the
carriers is raised above that of the lattice, to return to quasi-equilibrium the
carriers are cool down through carrier-phonon interactions. The major physical
process cause this mechanism are injected heating WL, stimulated
recombination, free carrier absorption and carrier energy relaxation. Both of
these effects result in raising the mean energy of the carrier distribution and thus
its temperature while the lattice temperature remains unchanged. After hundred
of femtoseconds the system is relax back to the thermal equilibrium and the hot
carrier is cool down through carrier-phonon scattering.
The large carrier density present in quantum wells and bulk can cause
carrier heating to be significant due to free carrier absorption. In quantum dots
however, the situation is more complicated. InAs dots grown on GaAs have a
large conduction-band offset. This, combined with the discrete energy spectrum
Chapter Three FWM and Pulse Propagation
30
reduces the carrier density at which gain is achieved. This in turn reduces the
free-carrier absorption and carrier heating effect [18]
3.3. Theory of pulse propagation in bulk SOA
Theory of pulse propagation in bulk material is well known for long time
[70], it treats the SOA as a two-level system where the transition occurs
between the conduction and valence bands. Let us assume that the propagation
of electromagnetic field inside the SOA is given by the following equation
22
2 20 (3.1)
EE
c t
c is the velocity of light, is the dielectric constant which is given by [71]
2 ( ) 3.2bn X N
bn is the background refractive index and ( )X N is the linear susceptibility
which represents the contribution of the charge carriers inside the active region
of the SOA. A simple model to represent the susceptibility is assumed to depend
on the carrier density )N( linearly and is given by [72]
( ) ( ) (3.3)N
p
nX N i g N
0 is frequency of the emitted photon, n is the effective mode index, ( )g N is
the optical gain approximately varies as [71]
0( ) (3.4)dg
g N N NdN
where dg
dN is the differential gain, Γ is the optical confinement factor, N is the
injected carrier density and 0N is the carrier density needed for transparency
and N is the linewidth enhanced factor which represent a coupling between the
phase and amplitude, the typical values of N , for bulk semiconductor is in
the range of (3-8) . The electric field associated with the optical pulse is given
as [71]
Chapter Three FWM and Pulse Propagation
31
0 0( )1ˆ( , , , ) ( , ) ( , ) . (3.5)
2
i k z tE x y z t F x y A z t e c c
where is the unit vector of polarization, ,F x y is the waveguide-mode
distribution, 00
nk
c
, and A(z, t) is the slowly varying amplitude of the
propagating wave. Substitute Eqs.(3.5-3.2), in Eq.(3.1), neglecting the second
derivatives of A(z, t) with respect to t and z, and integrating over the transverse
dimensions x and y, one obtains [71]
22 22 2
2 2 20 (3.6)o
b
d F d Fn n F
dx dy c
int
1 1(3.7)
2 2
o
g
idA dAXA A
dz dt nc
int is the loss coefficient, g is the group velocity ( / )g gc n , while gn is the
group refractive index. In bulk, the time evolution of carrier density (N)
describes by the following equation [71]
2( )(3.8)
c m
dN I N g NA
dt eV
Here V is the volume of the active region, c is the carrier lifetime, and m is
the cross section of the active region. The combination of Eq. (3.8) and
Eq.(3.4), gives [71]
20 (3.9)c sat
g gdg gA
dt E
0g is the small signal gain, which is given by [71]:
1 1 3.10o o oo o
c
dg I dg Ig N N
eVNdN dN I
where 0I is the current required for transparency.
3.11oo
c
eVNI
Chapter Three FWM and Pulse Propagation
32
satE is the saturation Energy, Eq. (3.7) can be further simplified by using the
retarded time frame[71]
(3.12)g
zt
Then, assume that i tA Pe and using Eq. (3.4), one obtain [71]
(1 ) 3.132
dA gi A
dz
0 3.14c Sat
g gdg gP
d E
int
1( ) 3.15
2
13.16
2
dPg P
dz
dg
dz
where P(z,τ) and φ(z, τ) are the instantaneous power and the phase of the
propagating pulse, respectively. The quantity sat sat cE P .
satP is the saturation
power of the amplifier which is given by [71].
3.17msat
c
Pa
The solution of Eqs. (3.14-3.16) generally requires some approximations,
if gint , it is possible to solve these equations in a closed form. In the
following, assume int =0. Eqs.(3.14-3.16) are then integrated over the amplifier
length to give [71].
( )( ) ( ) 3.18h
out inP P e
1
( ) ( ) ( ) 3.192
out in h
where inP and in are the power and phase of the input pulse. The function
h is defined by [71]
0
( , ) 3.20
L
h g z dz
Chapter Three FWM and Pulse Propagation
33
Physically, it represents the integrated gain at each point of the pulse
profile. If Eq. (3.14) is integrated over the length of amplifier and make use of
Eq. (3.18) to eliminate the product gP, the gain integral is the solution of the
following equation [71]
( )0 3.(
1)( )
1 2hin
c sat
g L h Pd he
d E
Numerically, For a given the gain (0g ) and input pulse shape inP (as an
example, consider a Gaussian pulse), Eq.(3.21) can be solved to obtain the gain
integral. The output pulse shape is then obtained from Eq.(3.18). Also,
Eq.(3.21) can be solved analytically. If the carrier lifetime c is much greater
than the input pulse width p the first term on the RHS of Eq. (3.21) can be
ignored. Physically this means that the pulse is so short that the gain has no time
to recover. Theoretically, the capture time of carrier for bulk semiconductor of
about (0.2-0.3 ns), and the above approximation is valid for p < 50 ps. In the
limit 1p
c
, the solution of Eq. (3.21) is [71]
where 0 0exp( )G g L is the unsaturated single-pass amplifier gain and ( )inU
represents the pulse energy which is given by the following equation[71]
( ) ( ) 3.23in inU P d
By definition, ( )in inU E , where inE is the input pulse energy and for Gaussian
pulse the solution of Eq. (3.23) is given by [71]
0
( ) 1 (3.24)2
inin
EU erf
where erf is the error function and τ0 is pulse width at half maximum.
( )
1( ) ln 1 1 3.22
in
sat
U
E
o
h eG
Chapter Three FWM and Pulse Propagation
34
3.4. integral gain and pulse propagation in QD SOA
In QDs, the definition of the integral gain must be different from that of the
bulk. Thus, the dynamics between (and inside) layers is different since there are
two-dimensional WL and a completely quantized QD layer. WL is considered
as a continuum state, compared to QD, due to its large number of states [73].
The transition between WL and QD takes a long time due to the difficulty of
energy conservation rules between WL and QD, and the phonon-bottleneck
effect arises [74]. Accordingly, these layers are included in our analysis to
obtain integral gain in QD structure. Because of the very few distance between
the hole levels due to their larger effective masses, a fast relaxation of the hole
is expected, and then, carrier dynamics are assumed to be limited by electrons in
the conduction band while holes are assumed to be in quasi-equilibrium at all
time in the valence band. This is a common assumption in the literature [75–
77]. Of course, calculation of the hole energy levels is included in the gain, a
static property, not a dynamic property of QDs. The dynamics in the QD SOA
are represented by 2-level rate equations [18]. From Eq. (2.16) and using
Eqs.(3.4, 3.10 and 3.11) the time evaluation of gain is derive as [20]
max
1 (3.25)2
QDo
c
g gdg g
d g
where
(3.26)QD
SHB
D dg
dN
1
1(3.27)w
SHB
e c
N
D
gmax is the maximum value of gain. SHB represents the rate at which the
quantum-dot ensemble will relax to thermal equilibrium via these capture and
escape dynamics. At low WL carrier densities, the relaxation is limited by how
quickly electrons can escape from the overly populated dots; however, as the
carrier density in the WL increases, it is the rate of carrier capture that limits the
Chapter Three FWM and Pulse Propagation
35
relaxation rate [18]. The integration of Eq.(3.25) over the amplifier length (L),
one obtain [20]
max
( )( ) ( )(3.28)
2
QDo
c
g L hd h hL
d g
3.5. FWM pulses
If we have two injected pulse signals (assuming transform-limited
Gaussian pulses) represented by [13]
2
0 0 2
0
1exp 3.29
2E A
2
1 1
1
1exp 3.30
2E A
where 0E is the pump signal at 0 , 1E is the probe signal at 1 , and is the delay
time between the two pulses. The FWHM of pump and probe pulses are 0 and
1 , respectively. The optical field at the input facet (z = 0) is [13]
0 10, 3.31i tE z t E E e
Note that is the phase between 0E and 1E . Integrating Eq. (3.31) yields [13]
2 1
( , ) (0, ) 3.32
h
i
out inE L E e
The small signal analysis can be applied to Eqs. (3.31) and (3.32), this
treatment leads to [13]
22 1
0 1 0( ) 1 ( ) 3.33
h
iE L e F E E
22 1
1 0 1( ) 1 ( ) 3.34
h
iE L e F E E
2 1 2 *
2 0 1( ) ( ) 3.35
h
iE L e F E E
Chapter Three FWM and Pulse Propagation
36
2,0
2
11 13.36
2 11
hx x
x SHB CH x
sat
ie iF C
iEi
E
In the expressions above, the field intensity,2
0E , is normalized to the
amplifier saturation intensity 2
satE , C is the phenomenological parameter to
compensate for the nonplanar nature of the waveguide [13]. Eq.(3.35) describes
the wave mixing product, E2, whose frequency is 2 0 , and is the gain
recovery time. Other products are also created but are all much smaller than E2
and therefore, they subsequently neglected. The set of Eqs. (3.33)-(3.35)
represents the relations of the three output fields to the two input fields. The
nature of these relations is determined by the function F , Eq. (3.36), which
contains all the physical details of the various nonlinearities. In Eq. (3.36), the
first term in the square brackets describes the wave mixing due to carrier density
pulsations (CDP). The second term was added to account for the summation of
three intraband processes. They are carrier heating of electrons in the
conduction band (CHc), of holes in the valence band (CHv), and spectral hole
burning (SHB). Each process has a characteristic time constant, x , a
corresponding nonlinear gain coefficient x , and a unique linewidth
enhancement factor x associated with it. The formula that describe travailing
the electric field in QD SOA can be written as [18]
2
0
2
1113.37
2 1
hQD x xN
x x
sat
iieF C
iE
E
where
Chapter Three FWM and Pulse Propagation
37
1
1 1 1 1 1 1
(3.38)
w w ws
e c c s e c s e c c
N N ND DD i i
D D
3.6. The nonlinear gain Coefficients
The nonlinear gain coefficient depend on the analytical solution of pulse
propagation inside QD SOA [3]. For the four-wave mixing contribution we
have; CDP, SHB and CH. In general, nonlinear gain coefficient due CDP is
assumed to be equal to unity [18]. Other nonlinear gain coefficients are
estimated from the normalized nonlinear susceptibility. The nonlinear gain
coefficients due SHB and CH are derived as
4
*
03
0
20
*0 1
0 02
(1 )
2( ) 1 ( ) ( )
( ) 21 ( ) ( )
(3.39)
SHB
SHB SHB CDP
Normalize
cv wj in j n j m
j cin
cv wrin
c
Xi
X
N
D
dg Nc ndN D
10, 0
, 2
0
(1 ) (3.40)1
w x x win xCH x CH
in s
N E Nhi E
K T i g
E is the energy difference between the chemical potential, the energy
needed to add one electron to the continuum, and the energy of an electron in a
quantum-dot bound state. *
em is as the effective mass for the electrons or holes
and is the effective height of the quantum-dot layer. hx (2 *
23
xk T m ) is the
heat capacity of the free electrons assuming a two-dimensional (2D) electron-
gas model [18].
Chapter Three FWM and Pulse Propagation
38
3.7. Wavelength Conversion in QD
Optical wavelength converters in semiconductor optical amplifiers have
become the key device of the future optical network and promise candidates for
future high-speed all-optical data routing applications [79]. In general, There are
three types of wavelength converters in SOA: cross gain modulation (XGM),
cross phase modulation (XPM), and recently, four-wave mixing (FWM), has
become one of the most preferred methods of wavelength conversion.
Unlike XGM and XPM wavelength converters, FWM preserves both the phase
and amplitude information. This is due to the non-changing nature of the
optical properties of the information signal during the conversion process
occurring within the SOA. The FWM-based wavelength converter in an
SOA presents a high bit rate capability up to tens of gigabits per second
(10Gb/s) [80].
The conversion efficiency of SOA, defined as the ratio between the power
of the converted signal at the device-output and the probe-power at the input
[76]. In QD SOA, FWM efficiency is given by [18]
2
0
2, 3.41
h
eff
sat
Ee F L
E
The Results and
Discussion
Chapter Four
Chapter Four The Theoretical Results
39
4.1. Introduction
Major parts of the current research in the natural and social sciences can
no longer be imagined without simulations, especially those implemented on a
computer, being a most effective methodological tool [81]. Simulating models
of the physical world is instrumental in advancing scientific knowledge and
developing technologies. Accordingly, the task has long been at the heart of
science [82]. Simulations are not always in dynamic models, where the
equations of the underlying dynamic model have time-varying coefficients.
This chapter involves the result of the theoretical simulation for QD
system composed of two-level rate equations depending on DMT and theory of
pulse propagation. Also the effect of CH has been involved in our analyses and
calculations. Most of FWM efficiency in QD SOA such as nonlinear gain
coefficients and linewidth due CDP, SHB and CH have been calculated
comparing with the others models [18,82,32,37]. This thesis not just about
theory of CH without the other mechanisms, it gives us the interaction between
them.
4.2. The Theoretical parameters
Our calculation has been performed a theoretical model to simulate the
influence of carrier heating in semiconductor optical amplifier compose of ten
layer of InAs QDs growth on 0.53 0.47In Ga As which was lattice matched to GaAs
and operating around 1.3 μm. The material parameters are *
em =0.023 0m , *
vm
=0.041 0m , g =0.345 eV, c 1 ps, s 0.2 ns, cv 150 fs, c =3.510
-22 m
-2,
0v , the amplifier length (L=3 μm), width (wd=20 μm), the thickness of each
layer is (Lw=10 nm), 0.027 and I=50 mA. These values are similar to those
used in [18,23], in most of the results to be presented, we consider a pump
wavelength of 1.33 μm, corresponding to a photon energy, 0 = 0.93 eV,
which is at the peak of the gain curve.
Chapter Four The Theoretical Results
40
4.3. Occupation probability of dot level
The occupation probability of Gs has been calculated by using Eq. (2.32),
where Fig.(4.1) shows the occupation probability of GS in the presence of CH
and without it. In this case the occupation probability increases with increasing
carrier density toward the saturation. The existing of CH effect reduces of
carrier density and therefore the occupation of carrier will be less. We believe
the reason behind this behavior is an interband times which are represented by
in . This time ( in ) is increase with existing CH relaxation time CH compare
with d in the model presented by [18], in previous model d represents the
time of SHB which is very fast compared to in .
Fig. (4.1): The GS occupation probability versus carrier density
Chapter Four The Theoretical Results
41
4.4. The gain and differential gain
There are several different physical mechanisms that can be used to
amplify a light signal, which correspond to the major types of optical amplifiers.
In SOA, stimulated emission in the amplifier's gain medium causes
amplification of incoming light where the electron-hole recombination occurs.
In QD system, there are several parameters that govern the production of gain
such as carrier density, relaxation time between dot and wetting layer, density of
state…etc. In this work, the calculation of gain based on fermi function that
expressed by Eq. (2.53). The effect of CH is present in our calculation, this is
obviously shown in Fig. (4.2). The material gain reduces with the existing of
CH effect, this result is due to the reduction in carrier density and occupation
probability in dot level (This is agree with [66]). Also the differential gain
respect to carrier density is determined in Fig. (4.3).
Fig.( 4.2 ): The gain versus wavelength
Chapter Four The Theoretical Results
42
Fig.( 4.3): Differential gain and linewidth enhancement factor versus wavelength
4.5. Transparency Carrier Density
The carrier needed for transparency represents a carrier density that
separates absorption from emission. Carrier density for transparency defines the
conditions of laser operation where the threshold of carrier density depends on.
Although the influence of heating effect on SOA and lasing operation have been
investigated intensively in bulk material but it didn’t take enough attention in
dot material. Fig. (4.4) shows the relation between the gain and carrier density,
we can see the carrier transparency is increase with inclusion of carrier heating
effect, this result is very important and it will be established predicts a new
ideas about laser action at specific thermal conditions.
Chapter Four The Theoretical Results
43
Fig.(4.4): The gain versus carrier density
4.6. Dynamic behavior in the time domain
Dynamic behavior of carrier density and occupation probability was also
studied. The numerical solution of rate equations (Eqs.(2.16 and 2.17 )) is given
by the following figures. Fig.(4.5) shows the solution of carrier density as a
function of time. With existence of carrier heating which is represented by the
term ( CH ), the carrier density is more gradient with CH than without heating
factor. Fig.(4.6) show the time series of occupation probability, with CH the
occupation probability is less, the reduction of with CH is the result of
decline of currier density. Fig. (4.6) can also provide an important information
about recovery time that limits the response of devices, with CH the increased
recovery time can give us an interpret of low efficiency.
Chapter Four The Theoretical Results
44
Fig.(4.5): The time domain of carrier density
Fig. (4.6): The occupation probability versus time
Chapter Four The Theoretical Results
45
4.7. Nonlinear gain coefficients
The most of nonlinear process in semiconductor are SHB and CH, the
nonlinear gain coefficients are one of best technique to studying this
phenomenon in SOA, so the nonlinear gain coefficients under study are:
4.7.1. Nonlinear gain coefficient due to SHB ( SHB )
Many of research submitted believed that the spectral hole burning
represents the major contributions in nonlinear gain compression [18]. SHB
process can be imagine by the creation of a hole in the gain spectrum due to
stimulated emission. SHB and CH cannot be separated because of the dynamic
of carriers, therefore we belief that to modeling a system to describe SHB
without contributing CH remains ineffective and comprehensive. In this work
we introduce a new description for simulating of SHB taking into count the
effect of CH. Fig.(4.7) shows the spectral hole burning versus carrier density
with contribution of CH (red dots) and the previous model [18] introduced by
(black dots) .
Fig.( 4.7): The nonlinear gain coefficient due SHB versus carrier density.
Chapter Four The Theoretical Results
46
4.7.2. Nonlinear gain coefficient due CH ( CH )
The effect of CH on the performance of SOA and laser is not less
importance than SHB. Although, there are a number of work have been reported
to studying carrier heating mechanism in bulk material, theory of CH in QD not
take enough attention. In this thesis, the simulation of CH in QD structure is
depended on nonlinear gain coefficient. Fig. (4.8) illustrate 3-dimensional plot
of CH WLN . CH is increase with increasing carrier density and reduced at
high detuning. at low detuning ( (1-10) GHz) the dependence of detuning is
very weak, with increasing of detuning (> 100 GHz) the dependence of
detuning becomes very reliance and the change with carrier density will be
clear. Fig. (4.9) show the effect of carrier time relaxation ( CH ) on CH . a
detuning (1GHz ), the increasing of CH will increase the value of CH . The
interpretation of this behavior lies in the value of time in which can be
considered a calibration of intraband relaxation time.
In Fig.(4.10),a comparison has been done between our model (QD) and
the bulk model in [18]. At low carrier density, the CH effect is completely
match, but with increasing of carrier density above the value 22 3( 10 )wN m , the
behavior of QD model becomes less than Bulk model. This figure gives us a
simple idea about a response of QD material to the heating effects.
Chapter Four The Theoretical Results
47
Fig. (4.8): 3-dimenssional plot of CH WLN .
Fig.( 4.9): The effect of CH on CH WLN curves
Chapter Four The Theoretical Results
48
Fig. (4.10): A comparison between QD model and the Bulk model .
4.8. Linewidth enhancement factor
Linewidth enhancement factor (LEF) is one of distinguishing features for
semiconductor amplifiers and lasers [83], or so called Hennerys factor. This
factor mainly reflects the material and design property of a laser and quantifies
the phase-amplitude coupling mechanism. -factor is nonzero value and leads
to many complex dynamics in semiconductor including linewidth broadening
and hence the -factor is referred to as the linewidth enhancement factor [84].
The typical value of LEF of about (2-7) in bulk material [85], it is less
than in QW and close to zero in QD [20]. The LEF has been calculated in
Fig.(4.3), it has a frequency dependence, at 1.33 μm the value of N is
about (1.13). Fig.(4.11.A) represents the value of LEF due to SHB. It is
constant (dose not dependent on carrier density and detuning). Fig. (4.11.B)
shows the behavior of LEF due to CH, it shows a strong detuning dependence.
The reason behind this dependence lies with the fast relaxation of SHB compare
relatively with slower relaxation of CH.
Chapter Four The Theoretical Results
49
Fig. (4.11.A): LEF due to SHB versus carrier density.
Fig.(4.11.B): LEF due to CH versus carrier density .
Chapter Four The Theoretical Results
50
4.9. The integral gain
The integral gain has been calculated using the analytical solution of
Eq.(3.28). With CH, the gain integral is less than these values without CH. This
reduction of integral gain can be interpreted by the decreasing of occupation
probability and carrier density with CH. In Fig. (4.12) the effect of CH at time
less than carrier heating relaxation time (≤ 2 ps) is ineffective, with time > CH ,
the effect of CH clearly appears.
Fig.(4.12): The time domain of integral gain .
4.10. The dynamic behavior and pulse effect
Theory of pulse propagation in QD has been employed in our theoretical
calculations, it has been focus on the impact of ultrafast Gaussian pulses on the
response of system because it is applicable in many experimental situations.
The change of pulse shape by the variation of full width at half maximum
(FWHM) clearly affect on the occupation probability (Fig.(4.13)) and carrier
density (Fig.(4.14)), and a result on the integral gain (Fig. (4.15)). Fig. (4.13)
Chapter Four The Theoretical Results
51
shown how GS recovery time increases with increasing the pulse width, this
feature can be exploit in improving the efficiency of devices.
Fig.(4.13): The pulse effect on occupation probability
Fig.(4.14): The pulse effect on the carrier density.
Chapter Four The Theoretical Results
52
Fig.(4.15) The pulse effect on the integral gain.
4.11. Wavelength conversion
FWM technique is a process by which optical signals at different (but
closely spaced) wavelengths mix to produce new signals at other wavelengths.
It has three different physical mechanisms contributing toward its conversion.
They are carrier density pulsation (CDP), spectral hole burning (SHB), and
carrier heating (CH) [32]. Figure (4.16) shows FWM and its components (CDP,
SHB and CH) respect to detuning. Although all components of FWM efficiency
exhibits a symmetric behavior, the total conversion efficiency shows an
asymmetric state.
Fig.(4.17) shows FWM efficiency for two-level system which is shown to
be a detuning independent expecting in the region ~ (50-300) GHz at negative
detuning. This is a required property in communication applications where
FWM is detuning independent. Also our results have also been compared with
experimental measurements done by Akiyama et. al. [86] for InAs/InGaAs/
Chapter Four The Theoretical Results
53
GaAs QD SOA as shown in Fig. (4.18). There was a good agreement with our
theoretical calculations
Fig.(4.16):Total FWM efficiency and its components versus detuning
Fig.(4.17): Total FWM efficiency versus detuning
Total
CDP
SHB
CH
Chapter Four The Theoretical Results
54
Fig.(4.18): Matching between the experimental (dot circles') measurements and our
calculations.
Conclusions
and
Future Works
Chapter Five
Chapter Five Conclusions and Future Work
55
Conclusions and Future Works
This chapter includes the summary of our study and the future works that
can be done depending on the concepts and principles presented so far.
5.1. Conclusions
The main conclusions of this study are summarized in the following points:
1. The inclusion of CH in QD theory made the values of gain, deferential
gain, linewidth enhancement factor, integral gain, occupation probability,
the nonlinear gain coefficient decrease, when the recovery time
increases.
2. The simulation of material gain in respect with carrier density proved that
the transparency carrier with CH is larger than that calculated in the Bulk
model.
3. The nonlinear gain coefficient due CH directly proportional to carrier
density and it shows strong dependence at high detuning, while SHB
nonlinear gain coefficient is inversely proportional to carrier density.
4. At low carrier density, the CH nonlinear gain coefficient has the same
behavior of bulk model, but with an increase in the carrier density above
the value (1022
m-3
), the behavior of QD model becomes less than bulk
model.
5. Linewidth enhancement factor due CH is less than that calculated in bulk
model, it is a detuning and carrier density dependence.
6. The effect of Gaussian pulse shape in time domain is presented through
the FWHM of input pulse, the occupation probability and carrier density,
the gain integral are decreased with the increasing of the FWHM, while
the recovery time increases with the increasing of the FWHM.
Chapter Five Conclusions and Future Work
56
7. All the components of FWM efficiency exhibit a symmetric behavior,
while the total conversion efficiency shows an asymmetric state, and in
the region ~ (50-300) GHz at negative detuning, the FWM efficiency
shows a detuning independence.
8. The theoretical results show a good agreement with the experiment.
5.2. Future Works
Semiconductor optical amplifiers are key devices in future
telecommunication networks for applications such as signal regeneration and
signal demultiplexing. The development of fabrication methods such as MBE
and MOCV and high nonlinearity of SOA make it attractive for use in
commercial optical communication systems. For these purposes the suggested
future studies are:
1. Investigation of CH contribution for 3-level rate equation and studying the
importance of inclusion of excited state in QD structure.
2. Simulation of the temperature of carriers in time domain. CH theories takes
it as an approximations from bulk
3. Examination of other conversion techniques such as cross-phase modulation,
cross-gain modulation.
4. Investigate the contribution of hole dynamic on the QD SOA theory.
References
References
57
References
[1] M. J. Connelly," Semiconductor Optical Amplifiers", Kluwer Academic
Publisher, 2002.
[2] B.H. Hong, “Modeling of Semiconductor Nanostructures: Electronic
Properties and Simulated Optical Spectra”, University of Hull, PhD thesis,
2011.
[3] T. W. Berg, S. Bischoff, I.Magnusdottir and J.Mørk, “Ultrafast Gain
Recovery and Modulation Limitations in Self-Assembled Quantum Dot
Devices”, IEEE Photonic Technology Letters, 13, 6, 541-534, 2001.
[4] P. Bhattacharya, K. K. Kamath, J. Singh, D. Klotzkin, J. Phillips, H.-
T.Jiang, N. Chervela, T. B. Norris, T. Sosnowski, J. Laskar, and M. R. Murty,
“In(Ga)As/GaAs self-organized quantum dot lasers: DC and small-signal
modulation properties”, IEEE Trans. Electron. Devices, 46, 871–883, 1999.
[5] D. Bimberg, N. Kirstaedter, N. N. Ledentsov, Z. I. Alferov, P. S. Kop’ev,
and V. M. Ustinov, “InGaAs–GaAs quantum-dot lasers,” IEEE J. Select. Topics
Quantum Electron., 3, 196–205, 1997.
[6] P. Borri, W. Langbein, J. M. Hvam, F. Heinrichsdorff, M.-H. Mao, and D.
Bimberg, “Ultrafast gain dynamics in InAs–InGaAs quantum-dot amplifiers,”
IEEE Photon. Technol. Lett., 12, 594–596, 2000.
[7] Y. Nambu and A. Tomita, “Spectral Hole Burning and Carrier-Heating
Effect on the Transient Optical Nonlinearity of Highly Carrier-Injected
Semiconductors”, IEEE Journal of Quantum Electronics, 30, 9, 1989-1994,
1994.
[8] J. Wang and H. Schweizer, “Nonlinear Gain Suppression in Quantum-Well
Lasers Due to Carrier Heating: The Roles of Carrier Energy Relaxation,
Electron-Hole Interaction, and Auger Effect”, IEEE Photonic Technology
Letters, 8,11, 1996.
References
58
[9] J. Singh, “Electronic and Optoelectronic Properties of Semiconductor
Structures”, Cambridge University Press, Book, USA, 109-114, 2003.
[10] Jin-Long Xiao, Yue-De Yang, and Yong-Zhen Huang, “Simulation on
Gain Recovery of Quantum Dot Semiconductor Optical Amplifiers by Rate
Equation”, IEEE,1, 4244-4180, 2009.
[11] Aaron J. Zilkie et al, “Carrier Dynamics of Quantum-Dot, Quantum-Dash,
and Quantum-Well Semiconductor Optical Amplifiers”, IEEE Journal of
quantum Elec., 43, 11, 982-991, 2007.
[12] Valentina Cesari , “Ultrafast carrier dynamics in p doped InGaAs/GaAs
quantum dots ”,Thesis ,Cardiff University, March 2009”,
[13] Niloy K. Dutta and Qiang Wang, “Semiconductor Optical Amplifiers”,
University of Connecticut, USA
[14] R. Mehra et al, “Optical computing with semiconductor optical amplifiers”,
Optical Engineering, 51, 8, 0809011-080909, 2012.
[15] V. von, “Nonlinear Applications of Semiconductor Optical Amplifiers for
All-Optical Networks”, Thesis, Berlin University, 2007.
[16] D. A. Clugston and P. A. Basore, “Modelling Free-carrier Absorption in
Solar Cells”, Progress in photovoltaics: research and applications, 5, 229-236 ,
1997.
[17] J. Lamperski, “Cross-gain Modulation Techniques for All Optical
Wavelength Conversion”, XIV Poznań Telecommunications Workshop, Poland,
2010.
[18] D. Nielsen and S. L. Chuang, “Four-Wave Mixing and Wavelength
Conversion in Quantum Dot”, Physical Review B 81, 2010.
[19] A. Rostami, H. Baghban, R. Maram, “Nanostructure Semiconductor
Optical Amplifiers”, springer, Book, 2011.
[20] A.H. Flayyih,” Four-Wave Mixing in Quantum Dot Optical Amplifier”,
PhD thesis, 2013.
References
59
[21] M. Willatzen, T. Takahashi, and Y. Arakawa, “Nonlinear Gain Effects Due
to Carrier Heating and Spectral Hole burning in Strained-Quantum- Well
Lasers”, IEEE Transactions Photonics Technology Letters, 4, 7, 1992.
[22] F. Jahnke and S. W. Koch, “Theory of carrier heating through injection
pumping and lasing in semiconductor microcavity lasers”, Optical Society of
America, Optics letters, 18, 17, 1438-1440, 1993.
[23] A. Uskov, J. Mørk, and J. Mark, “Wave Mixing in Semiconductor Laser
Amplifiers Due to Carrier Heating and Spectral-Hole Burning “,IEEE Journal
Quantum Electronics, 30, 1769-1781, 1994.
[24] C. Tasi et al, ”Effects of Hot Phonons on Carrier Heating in Quantum Well
Lasers ”, IEEE Photonics Technology, 7, 9, 950-952, 1995.
[25] V. Uskov, J. R. Karin, J. E. Bowers, J. G. McInerney, and J. Le Bihan,
“Effects of Carrier Cooling and Carrier Heating in Saturation Dynamics and
Pulse Propagation Through Bulk Semiconductor Absorbers”, IEEE Journal
Quantum Electronics, 34, 11, 2162-2171, 1998.
[26] J. Wang and H. C. Schweizer, “A Quantitative Comparison of the Classical
Rate-Equation Model with the Carrier Heating Model on Dynamics of the
Quantum-Well Laser: The Role of Carrier Energy Relaxation, Electron–Hole
Interaction, and Auger Effect”, IEEE Journal Quantum Electronics, 33, 8, 1350-
1359, 1997.
[27] T. V. Sarkisyan, A. N. Oraevsky, A. T. Rosenberger, R. L. Rolleigh and D.
K. Bandy, “Nonlinear gain and carrier temperature dynamics in semiconductor
laser media”, Optical Society of America B, 15, 3, 1107-1119, 1998.
[28] X. Yang , D. Lenstra, G.D. Khoe and H. Dorren, “Nonlinear polarization
rotation induced by ultrashort optical pulses in a semiconductor optical
amplifier”, Optics Communications, 223, 169–179, 2003.
References
60
[29] Y. Ben-Ezra, B. I. Lembrikov, and M. Haridim, “Acceleration of Gain
Recovery and Dynamics of Electrons in QD-SOA”,IEEE Journal Quantum
Electronics ,41, 10, 1268-1273, 2005.
[30] O. Qasaimeh, “Novel Closed-Form Model for Multiple-State Quantum-Dot
Semiconductor Optical Amplifiers”, IEEE Journal Quantum Electronics, 40, 7,
652-657, 2008.
[31] N. Majer, K. Lüdge, J. Gomis-Bresco, S.Dommers, U.Woggon et al.,
“Impact of carrier-carrier scattering and carrier heating on pulse train dynamics
of quantum dot semiconductor optical amplifiers”, Applied Physic Letters, 99,
1311021-1311023, 2011.
[32] A. H. Flayyih and A. H. Al-Khursan, “Four-wave mixing in quantum dot
semiconductor optical amplifiers”, OSA, Applied Optics D, 52, 14, 3156-3166,
2013.
[33] A. H. Flayyih and A. H. Al-Khursan, “Integral gain in quantum dot
semiconductor optical amplifiers” , Superlattices and Microstructures, 62, 81–
87, 2013.
[34] A. H. Flayyih and A. H. Al-Khursan, “Carrier heating in quantum dot
structures”, Modern Physics Letters B, 28, 1450032-145039, 2014
[35]Chin-Yi Tsai, Chin-Yao Tsai, Robert M. Spencer, Yu-Hwa Lo, “Nonlinear
Gain Coefficients in Semiconductor Lasers: Effects of Carrier Heating”, IEEE
Journal of Quantum Electronics, Vol. 32, No. 2, PP. 201-212, 1996.
[36] M. Willatzen, A. Uskov, J. Mork, H. Olesen, B. Tromborg, and A. P.
Jauho, “Nonlinear gain suppression in semiconductor lasers due to carrier
heating,” IEEE Photon. Technol. Lett., Vol. 3, PP. 606609, 1991.
[37] Alexander V. Uskov, Christian Meuer, Holger Schmeckebier, and Dieter
bimberg, “Auger Capture Induced Carrier Heating in Quantum Dot Lasers and
Amplifiers”, Applied Physics Express Vol. 4, PP. 022202, 2011.
References
61
[38] Chin-Yi Tsai, Yu-Hwa Lo, Robert M. Spencer, Chin-Yao Tsai, “Effect of
hot phonon in QW Lasers”, IEEE Photonics Technology letters, Vol. 7, No. 9,
PP. 950-952, 1995.
[39] J. Siegret, "Carrier dynamics in semiconductor quantum dots", PHD
Thesis, Sweden, 2006.
[40] B. Zaho, T. R. Chen and A. Yariv, “On the high speed modulation
bandwidth of quantum well lasers”, Applied Physics Letters, Vol. 60, PP. 313,
1992.
[41] Sidney C. Kan, Dan Vassiloveski, Ta. C. Wu and Kam Lau, “Quantum
capture limited modulation bandwidth of quantum well, wire, and dot lasers ”,
Applied Physics Letters, Vol. 62, PP. 2307, 1993.
[42] O. Qasaimeh, “Ultrafast dynamic properties of quantum dot
semiconductor optical amplifiers including line broadening and carrier heating”,
IEE Proc.-Optoelectronics, Vol. 151, No. 4, PP. 205-210, 2004
[43] S. L. Chuang, “Physics of Optoelectronic Devices”, New York, NY: John
Wiley and Sons, Inc., Book, 1995.
[44] R. J. Boyd, “Nonlinear Optics”, New York, October, 2007
[45] H. Haug and S. W. Koch, “Quantum Theory of the Optical and Electronic
Properties of Semiconductors”, Singapore: World Scientific, 1990.
[46]M. Sargent, S. W. Koch and W. W. Chow, “Side Mode Gain in
Semiconductor Lasers,” Optical Society of America , 9, 1288-1298, 1992.
[47] V. Von, “GaAs-Based Semiconductor Optical Amplifiers with Quantum
Dots as an Active Medium”, Diplom- Physiker, Berlin, 2007.
[48] M. Sugawara, T. Akiyama, N.Hatori, Y. Nakata, H.Ebe and H. Ishikawa
“Quantum Dot Semiconductor Optical Amplifiers for High-bit-rate Signal
Processing up to 160 Gb/sec and a New Scheme of 3R Regenerators”,
Measurement Science and Technology, 13, 1683–1691,2002.
References
62
[49] J. P.Turkiewicz, ”Application of O-band Semiconductor Optical Amplifier
an Fiber-Optic Telecommunication Networks
[50] S. Barua, N. Das, S. Nordholm and M. Razaghi, “Analysis of Nonlinear
Pulse Propagation Characteristic in Semiconductor Optical Amplifier for
different Pulse Shapes ”, International Journal of Electrical, computer and
communication, 9, 1, 16-20, 2015.
[51] C.Politi, D.Klonidis, andM. J. O’Mahony, "Wave band converters based on
four-wave mixing in SOAs", IEEEJ. Light wave Technol.,24, 1203-1217, 2006.
[52] H. Su, H. Li, L. Zhang, Z. Zou, A. L. Gray, R. Wang, P. M. Varangis, and
L. F. Lester, "Nondegenerate four-wave mixing in quantum dot distributed
feedback lasers", IEEE Phot. Technol. Lett., 17,1686-1688, 2005.
[53] T. Kita, T. Maeda, and Y. Harada, "Carrier dynamics of the intermediate
state in InAs/GaAs quantum dots coupled in a photonic cavity under two-
photon excitation", Phys. Rev. B 86, 2012, 035301.
[54] C. Wang, F. Grillot, and J. Even, " Impacts of wetting layer and excited
state on the modulation response of quantum-dot lasers", IEEEJ. Quantum
Electron., 48, 1144-1150, 2012.
[55] B. Al-Nashy and Amin H. Al-Khursan, "Completely inhomogeneous
density-matrix theory for quantum-dots”, Optical and Quantum Electronics41,
989-995, 2010.
[56] C.Politi, D.Klonidis, and M. J. O’Mahony, "Dynamic behavior of
wavelength converters based on FWM in SOAs," IEEEJ. Quantum
Electron.,42,108-125, 2006.
[57] A. Mecozzi, S . Scotti, A. D’Ottavi, E. Iannone, and P. Spano, "Four-
wave mixing in traveling-wave semiconductor amplifiers", IEEEJ. Quantum
Electron.,31, 689-699, 1995.
[58] J. Liu and T. B. Simpson, "Four-wave mixing and optical modulation in
a semiconductor laser", IEEEJ. Quantum Electron.,30, 951-965, 1994.
References
63
[59] T. Mukai and T. Saitoh, "Detuning characteristics and conversion
efficiency of nearly degenerate four-wave mixing in a 1.5 m traveling-wave
semiconductor laser amplifier", IEEEJ. Quantum Electron.,26, 865-875, 1990.
[60] Amin H. Al-Khursan, M. K. Al-Khakani, K. H. Al-Mossawi, "Third-order
non-linear susceptibility in a three-level QD system", Photonics and
Nanostructures-Fundamentals and Applications, 7, 153–160, 2009.
[61] J. Kim, C.Meuer, D.Bimberg, and G. Eisenstein, "Numerical simulation of
temporal and spectral variation of gain and phase recovery in quantum-dot
semiconductor optical amplifiers", IEEEJ. Quantum Electron.,46, 405-413,
2010.
[62] Amin H. Al-Khursan, “Intensity noise characteristics in quantum-dot
lasers: four-level rate equations analysis”, Journal of Luminescence, 113, 129-
136, 2005.
[63] I. O’Driscoll, P. Blood, and Peter M. Smowton,"Random population of
quantum dotsin InAs–GaAs laser structures," IEEEJ. Quantum Electron.,46,
525-532, 2010.
[64] M. Sugawara , T. Akiyama, N. Hatori, Y. Nakata, K. Otsubo, and H.
Ebe,"Quantum-dot semiconductor optical amplifiers", Proceedings of SPIE
4905, 259-275, 2002.
[65] J. Kim, M. Laemmlin, C. Meuer, D. Bimberg, and G. Eisenstein, "Static
gain saturation model of quantum-dot semiconductor optical amplifiers", IEEEJ.
Quantum Electron.,44,658-666, 2008.
[66] M. Xia and G. Shiraz "Analysis of Carrier Heating Effect of Quantum Well
Semiconductor Optical Amplifiers Considering Holes Non-Parabolic density of
state", Optical Quantum Electron, 11082-14, 2014.
[67] A. H. Flayyih and A. H. Al-Khursan, ”Theory of four-wave mixing in
quantum dot semiconductor optical amplifiers”, Journal of Physics (IOP) D:
Applied Physics, 46, 445102, 2013.
References
64
[68] N. A. Ansari, “Studying of The Theory of Four-Wave Mixing ”, PhD
Thesis, Quaid -i- Azam Unversity, Pakistan, 1989.
[69] M.C.C.Lakare,”WavelengthConversionByUsinga1550nmLaserAmplifier”,
MSC Thesis, Eindhoven University Technology, 1999.
[70] M. L. Frantz and J. N. Nodvik, “Theory of Pulse Propagation in a Laser
Amplifier”, Journal of Applied Physics, 34, 8, 2346-2349, 1963.
[71] G. P. Agrawal and N. A. Olsson, “Self phase modulation and spectral
broadening of optical pulses in semiconductor laser amplifiers”, IEEE J.
Quantum Electron., 25,2297–2306, 1989.
[72] G. P. Agrawal and N. K. Dutta, “Long- Wavelength Semiconductor
Laser” New York: Van Nostrand Reinhold, Ch2, 1986.
[73] T.W. Berg, J. Mørk, “Saturation and noise properties of quantum-dot
optical amplifiers”, IEEE J. Quantum Electron. 40 1527–1539, 2004.
[74] Amin H. Al-Khursan, Intensity noise characteristics in quantum-dot lasers:
four-level rate equations analysis, 113, 129–136, 2005
[75] A. Markus, M. Rossetti, V. Calligari, D. Chek-Al-Kar, J.X. Chen, A. Fiore,
R. Scollo, “Two-state switching and dynamics inquantum dot two-section
lasers”, J. Appl. Phys. 100, 113104, 2006.
[76] A. Matsumoto, K. Kuwata, A. Matsushita, K. Akahane, K. Utaka,
“Numerical analysis of ultrafast performances of all-opticallogic-gate devices
integrated with InAs QD-SOA and ring resonators”, IEEE J. Quantum
Electronics 49, 51–58, 2013.
[77] H. Simos, M. Rossetti, C. Simos, C. Mesaritakis, T. Xu, P. Bardella, I.
Montrosset, D. Syvridis, “Numerical analysis of passively mode-locked
quantum-dot lasers with absorber section at the low-reflectivity output facet”,
IEEE J. Quantum Electron. 49, 3–10, 2013.
References
65
[78] M. Shtaif and G. Eisenstein, “Analytical Solution of Wave Mixing between
Short Optical Pulses in a Semiconductor Optical Amplifier”, Applied Physics
Letters, Vol. 66, No. 12, PP. 458–1460, 1995
[79] J. Mørk and A. Mecozz, “Theory of Nondegenerate Four-Wave Mixing
Between Pulses in a Semiconductor Waveguide”, IEEE Journal of Quantum
Electronics, 33, 4, 545-555, 1997.
[80] A. Kaur, K. Singh and B.Utreja, “Performance analysis of semiconductor
optical amplifier using four wave mixing based wavelength Converter for all
Optical networks”, International Journal of Engineering Research and
Applications (IJERA), 3, 4, 108-113, 2013.
[81] R. Kippenhahn and A. Weigert. “Stellar Structure and Evolution”,
Springer, Berlin, 1991.
[82] T. Johnson et al, “what is quantum simulation”, Centre for Quantum
Technologies, National University, 2014
of Singapore
[83] C. Henry, “Theory of linewidth of semiconductor laser”, IEEE, Journal of
Quantum Electronics, 18, 259-264,1982.
[84] M. J. Khudhair, “ Chaos synchronization of optical communication system
using a laser diode subjected to double optical feedback ”, MSC thesis, Al-
Mustasiriyah Un., Iraq, 2015.
[85] M. Osinski, J. Buus, “Linewidth Broadening Factor in Semiconductor
Lasers- An Overview”, IEEE Journal of Quantum Electronics, 23, 9-29 , 1987.
[86] T. Akiyama, H. Kuwatsuka, N. Hatori, Y. Nakata, H. Ebe and M.
Sugawara, "Symmetric Highly Efficient (~0dB) Wavelength Conversion Based
on Four-Wave Mixing in Quantum Dot Optical Amplifiers", IEEE Photonics
Technology Letters, 14, 1139, 2002.
الخالصة
الخالصةأ ظش٠ح زشاسج اساالخ ف اضخاخ اثصش٠ح شثح اصح اا٠ح ذأخز االرا
اشافمح ع اشغ عادح اظاشوالس١ى١ح اارج اظش٠ح ع ارج أوثشاعرذخ إر اىاف،
طش٠مح و١ح.ترأث١شاخ اسشاس٠ح
زشاسج اساالخ ف اضخ شثح اص ذأث١شخذ٠ذ ذساسح ص١غحف ز اذساسح، ذ ذمذ٠
اصففح ظش٠ح سش٠ا اثضاخ ظش٠ح وثافحتاعراد ظا ؤف سر١٠ اا اى
امص١شج ف ااد شثح اصح. أ دساسح ز اظاشج ذد خالي عا اشتر االخط از
تاالعراد ز١ث ذ زساب عا اشتر االخط ٠عرثش أفض اطشق ف دساسح اظاش االخط١ح،
عا اشتر االخط ااذح ارأث١ش اسشاس إ سش٠ا اثضاخ امص١شج.ارس١ ع ارج
ساالخ ذ دساسر ماسر ع ارج اىالس١ى، لذ خذ أ ارج امرشذ ٠طثك ع ارج
10)) سا٠ح ا اىالس١ى عذا ذى وثافح اساالخ21
-1022
)m-3
، ا ص٠ادج عذد اساالخ فق (
10ام١ح )22
m-3
زا . وزه سن ارج اى س١ى أوثش اخفاضا اسن اىالس١ى( فا
ااذح ع زشق اشتر االخطعا اخفاض ف ازرا١ح اإلشغاي وثافح اساالخ إد ارأث١ش ٠ؤ
ارائح ار زصا .ارأث١شاسرعادج اشتر تخد زا ، ت١ا زع ص٠ادج ف ص اساالخ اط١ف١ح
اثضاخ ص٠ادج عشض أ ز١ث صف اشذج،رذأث١ش شى اثضح ارث تاعشض عذ ع١ا
ازرا١ح اإلشغاي وثافح اساالخ عا اشتر ارفاض طشد٠ا ع ص اسرعادج ع ا ٠راسة عىس١
اشتر.
اسشاس ساالخ إعادج ص١اغح ظش٠ح ضج األاج األستعح رج اى رأث١شاإ
( از inارفاع ت١ ز اظاشج اظاش األخش از ظش تشى اضر خالي اض ) خالي
ساسح وفاءج ٠ث اض اؤثش ع١اخ اى١ح. ع ضء ارج ادذ٠ذ ضج األاج األستعح ذ د
ارس٠ اراظش اساص ت١ شوثاذا، لذ أظشخ ارائح اظش٠ح ذطاتما خ١ذا ع رائح ع١ح شسج
.ف دالخ عا١ح
جمهورية العراق وزارة التعميم العالي و البحث العممي
جامعة ذي قار كمية العموم
تأثير تسخين الحامالت في المضخم البصري لشبه
موصل نقطي كمي
مجمس مقدمة إلى رسالة ذي قارجامعة –كمية العموم
الماجستيرجزء من متطمبات نيل درجة و هي الفيزياء في عموم
من قبل
سالم ثامر جمود
بأشراف أحمد حمود فميح د.أ.م. فالح حسن حنون د.أ.م.
م5102 ه 1437
ج
ا
م
ع
ة
ذ
ي
ق
ا
ر
ك
ل
ي
ة
ا
ل
ع
ل
و
م
C
H
3
H