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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

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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


Date: - -2015

Signature:

Name: Dr. Emad Abd uL-Razaq Salman


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].


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


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] .


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].


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].


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


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


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


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.


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


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


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


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]


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


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].


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].


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


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]


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


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


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]


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


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


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


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


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


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


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]


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


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


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


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


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


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


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].


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.


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


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

https://en.wikipedia.org/wiki/Stimulated_emission

https://en.wikipedia.org/wiki/Gain_medium

https://en.wikipedia.org/wiki/Electron

https://en.wikipedia.org/wiki/Electron_hole

https://en.wikipedia.org/wiki/Carrier_generation_and_recombination


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.


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.


44

Fig.(4.5): The time domain of carrier density

Fig. (4.6): The occupation probability versus time


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.


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.


47

Fig. (4.8): 3-dimenssional plot of CH WLN .

Fig.( 4.9): The effect of CH on CH WLN curves


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.


49

Fig. (4.11.A): LEF due to SHB versus carrier density.

Fig.(4.11.B): LEF due to CH versus carrier density .


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)


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.


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/


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


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

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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

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، ا ص٠ادج عذد اساالخ فق (

10ام١ح )22

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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.

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