ك ج Carrier Heating Effects in Quantum Dot Semiconductor Optical Amplifier

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1 Republic of Iraq Ministry of Higher Education And Scientific Research Thi-Qar University College of Science H C ك ج 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 (Assistant Professor) Dr. Ahmed H. Flayyih (Assistant Professor) 15 A.D A. H

2 ب س م ا ه لل ال هر ح ن ال هرح ي ق ال وا س ب ح ان ك إ ل ل ن ا ع ل م ل ال ع ل يم أ ن ت إ ن ك ع ل م ت ن ا م ا ال ح ك يم صدق اهلل العظيم سورة البقرة: اآليت 3

3 Dedication I dedicate my humble effort TO My father's spirit My mother My brothers and sisters

4 Examination Committee Certificate 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

5 Supervisor Certification 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. Signature: Name: Dr. Falah H. Al-asady Title: Assistant Professor Date: Signature: Name: Dr. Ahmed H. Flayyih Title: Assistant Professor Date: In view of the available recommendations, we forward this thesis for discussion by the examining committee. Signature: Name: Dr. Emad Abd ul-razaq Salman Title: Assistant Professor Head of the Physics department Date: - -15

6 Acknowledgements 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 I

7 Abstract 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 ((1 1-1 ) m -3 ) for carrier density, but with the increasing of carriers above the value (1 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. II

8 Abstract 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. III

9 List of Symbols List of Symbols Symbols A(z, t) a n c C c D dg dn E E 1 E E t E in E sat E F x, y f c L f c g g SHB g g CDP CH g g H H H max h I I CH SHB M m N * e w Definition slowly varying amplitude of the propagating wave absorption renormalized for the occupation probability the phenomenological parameter to compensate for the nonplanar nature of the waveguide the velocity of light total number of states the differential gain electrical intensity of the pump electrical intensity of the probe electrical intensity of conjugate formed through nonlinear mixing electric field of the interacting light the input pulse energy the saturation energy the energy difference between the chemical potential the waveguide-mode distribution function fermi function fermi function at lattice temperature material gain Gain due to spectral hole burning Nonlinear gain due to carrier density pulsation Nonlinear gain due to carrier heating the maximum value of gain the small signal gain Hamiltonian operator of the system Hamiltonian for a free atom without perturbation Hamiltonian at interaction (the interaction of the atom with the field of applied radiation) The gain integral Blanck Constants injected current the current required for transparency The nonlinear gain coefficient due to carrier heating The nonlinear gain coefficient due to spectral hole burning. the effective height of the quantum-dot layer the electric dipole moment operator of the atom the effective mass for the electrons carriers density IV

10 List of Symbols Symbols n g N n b n n Pz (, ) Pin P s t s e c s CH cv 1, SHB U U in () in g V 1 i X( N ) χ( ) nm N ˆ int m in, ( z, ) Definition the grope refractive index the carrier transparency the background refractive index the refractive index the effective mode index the instantaneous power of the propagating pulse The power of the input pulse the saturation power of the amplifier the carrier lifetime carrier escape time carrier capture time spontaneous time carrier heating time constant decoherent time The full width at half maximum of pump and probe pulses the spectral hole time constant. total intraband time constant the pulse energy The energy density the grope velocity the volume of the active region angular frequency of the pump angular frequency of the probe angular frequency of the conjugate signal transition frequency the linear susceptibility Lorentzian lineshape decay rate the optical confinement factor the linewidth enhanced factor the loss coefficient the unit vector of polarization the permittivity of free space the dielectric constant the cross section of the active region The phase of the input pulse the phase of the propagating pulse V

11 List of Symbols Symbols n nm c c, v cv, j Definition the delay time between the two pulses the energy eigen functions the detuning Four wave mixing conversion efficiency density operator density matrix of motion occupation probability of ground state Electron and hole occupation probabilities in the valence and conduction band. dipole term wave function VI

12 List Of Abbreviations List of Abbreviations Attribute CH CDP FCA FWM FWHM GaAs GS LO LEF MBE MOCVD QD QW QD SOAs SHB SOA TPA WC WL XGM XPM Meaning carrier heating carrier density pulsation free carrier absorption four wave mixing full width at half maximum gallium arsenide ground state longitudinal optical Linewidth enhancement factor molecular beam epitaxial metal-organic chemical-vapour deposition Quantum dot quantum-well Quantum dot semiconductor optical amplifiers spectral hole burning Semiconductor Optical Amplifier two-photon absorption wavelength conversion Wetting layer cross gain modulation cross phase modulation VII

13 List Of Figures List of Figures Fig. No. (1.1) (1.) (1.3) (1.4) (.1) (.) (.3) (4.1( (4.) (4.3) (4.4) (4.5) (4.6) (4.7) (4.8) (4.9) (4.1) (4.11.A) (4.11.B) (4.1) (4.13) (4.14) )4.15( )4.16( (4.17) (4.18) Description The Semiconductor optical amplifier schematic diagram Semiconductor materials used in laser fabrication at different regions of the spectrum the SOA structure Temporal evolution of carrier distribution after exciting by short optical pulses Diagram of carrier relaxation processes in QD Semiconductor band structure Optical field of pump, probe and conjugate versus frequency the GS occupation probability versus carrier density the gain versus wavelength differential gain and linewidth enhancement factor versus wavelength the gain versus carrier density the time domain of carrier density The occupation probability versus time The nonlinear gain coefficient due SHB versus carrier density 3-dimenssional plot of CH NWL. the effect of CH on CH NWL curves A comparison between QD model and the Bulk model LEF due SHB versus carrier density LEF due CH versus carrier density the time domain of gain integral the pulse effect on occupation probability the pulse effect on the carrier density the pulse effect on the gain integral Total FWM efficiency and its components versus detuning Total FWM efficiency versus detuning Matching between the experimental measurements and our calculation Page VIII

14 Contents Contents Subjects Acknowledgements Abstract List of Symbols List of Abbreviations List of Figures Contents Chapter One: Introduction 1.1. Introduction 1.. Semiconductor Optical Amplifiers 1.3. Quantum Dot Semiconductor Optical Amplifiers 1.4. SOA Material 1.5. SOA Structure 1.6. Dynamic Recovery in SOA 1.7. SOA Nonlinearities 1.8. SOA Gain 1.9. Literature Review 1.1. The Aim of This Work Chapter Two: Theory of CH in QD SOA.1.Introduction..Density Matrix Theory.3.The Rate Equations.4.Theory of Nonlinear Process in QD Chapter Three: FWM and Pulse propagation 3.1.Introduction 3..Four-Wave Mixing in semiconductor Pages I II IV VII VIII IX IX

15 Contents 3.3.Theory of Pulse propagation in Bulk SOA 3.4.Gain integral and pulse propagation in QD SOA 3.5.FWM pulses 3.6.The nonlinear Gain Coefficients 3.7. Wavelength Conversion in QD Chapter Four : The Theoretical Results 4.1. Introduction 4.. The Theoretical parameters 4.3. Occupation probability of dot level 4.4. The gain and differential gain 4.5. Transparency Carrier 4.6. Dynamic behavior in the time domain 4.7. Nonlinear gain coefficients 4.8.Linewidth enhancement factor 4.9. The gain integral 4.1.The dynamic behavior and pulse effect 4.11.Wavelength conversion Chapter Five : Conclusions and Future Works 5.1. Conclusions 5.. Future Works References X

16 Chapter One Introduction

17 Chapter One Semiconductor Optical Amplifier 1.1. Introduction The first research on semiconductor optical amplifiers (SOAs) has been begun in the 196s at the time of the invention of laser. The early devices were based on gallium arsenide operating at low temperatures [1]. In the late 196s, 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 []. 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 1 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]. 1

18 Chapter One Semiconductor Optical Amplifier 1.. 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] 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 [1]. 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

19 Chapter One Semiconductor Optical Amplifier 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. [1] 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.). Many of these material systems are ternary (three elements) and quaternary (fourelement) 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.): Semiconductor materials used in laser fabrication at different regions of the spectrum [13]. 3

20 Chapter One Semiconductor Optical Amplifier 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 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]. 4

21 Chapter One Semiconductor Optical Amplifier 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]. 5

22 Chapter One Semiconductor Optical Amplifier 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] 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] g g g g CDP SHB CH Interband Intraband (1.1) where g is total gain, g CDP 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 g CH and g SHB 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 6

23 Energ y Chapter One Semiconductor Optical Amplifier 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. Carrier Density SHB Heating Cooling Electrical Pumping Fig.(1.4) Temporal evolution of carrier distribution after exciting by short optical pulses [] 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 ([1], 199): 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 7

24 Chapter One Semiconductor Optical Amplifier important mechanism for nonlinear gain coefficient and its effect increased faster with strain. F. Jahnke and S. W. Koch ([], 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 ([3], 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 ([4]. 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 ([5], 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([6], 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 8

25 Chapter One Semiconductor Optical Amplifier 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 ([7], 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 ([8], 3): 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 ([9], 5): 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 ([3], 8): 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], 1): This study introduced an analytical model to determine FWM in QDs base on density-matrix formalization for single boundstate 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], 11): 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. 9

26 Chapter One Semiconductor Optical Amplifier A.H. Al-Khursan et al ([3], 13): 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], 13): 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], 14): 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 The aim of this work Carrier heating phenomenon in semiconductor materials has been known in the late of 198 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, ]). So the aim of this thesis is introduced a new 1

27 Chapter One Semiconductor Optical Amplifier 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.. 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. 11

28 Chapter Two Theory of Carrier Heating in Quantum Dot Semiconductor Optical Amplifier

29 Chapter Two Theory of CH in QD SOA.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 [35-36,6], 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 [3], 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 [18,3,4-41]. Carrier heating influence on the performance of lasers and amplifiers in bulk and QW has been reported by a number of work [31,3,4], 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)... 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 1

30 Chapter Two Theory of CH in QD SOA 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] (.1) Where is the wave function which obeys the Schrödinger equation [43] H iħ t (.) where H denotes the Hamiltonian operator of the system. We assume that H can be represented as [43] H H H (.3) where H 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 H. It is usually assumed that this interaction energy is given as [43] H M. E t.4 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 H and thus satisfy the equation H n E n n. As a consequence, the matrix representation of H is diagonal that is [44], H, nm Ennm.5 The commutator can be expanded and the summation over ν can be performed formally to write as[43]. H, ( H H ) ( H H ) nm nm, nv vm nv, vm v v ( E E ) n nv vm nv vm m E E.6 n m nm 13

31 Chapter Two Theory of CH in QD SOA The transition frequency (in angular frequency units) is nm E n E m.7 The density matrix equation of motion with the phenomenological inclusion of damping is given by [43] i eq H,.8 nm nm nm nm nm Substituting Eqs. (.3,.6 and.7) in Eq.(.8), the equation of motion is i i eq H,.9 nm nm nm nm nm nm nm The expanded of as a linear combination [44]... (.1) () 1 (1) () nm nm nm nm The solutions of Eq.(.9) are [44] i () () () ( ) nm nm nm nm nm nm eq.1a (1) nm ( ) nm (1) 1 () i i H,.1B nm nm nm () 1 () i i H,.1C nm nm nm Equation (.1A) describes the time evolution of the system in the absence of any external field. We take the steady-state solution to this equation to [44] where ( eq ) (), Now that is known, Eq. (.1B) can be () ( eq ) nm nm nm (1) integrated. To do so, we make a change of variables by representing nm as [44] i t (1) (1) ( )e nm nm t S t.11 nm nm The derivative of (1) nm can be represented in terms of nm i t i t (1) (1) (1) nm i nm nm S nm t S nm t (1) S nm as [44] nm nm nm nm ( )e ( )e.1 These forms are substituted into Eq. (.1B), which then becomes [44] 14

32 Chapter Two Theory of CH in QD SOA S i inm nm t H, e.13 (1) () nm nm This equation can be integrated to give [44] t (1) i t S nm H t () inm nm ( ), e dt.14 nm This expression is now substituted back into Eq. (.11) to obtain [44] (1) nm i ( ), e t H t dt.15 t () inm nm ( t t ) nm.3. The Rate Equations The model represents carrier dynamics in two-level system (WL, GS) can be seen in Fig.(.1) τ c τ e CB GS (QD Level) VB Fig. (.1) Diagram of carrier relaxation processes in QD [18]. The rate equations describe dynamic of carriers for QD -level system including CH contribution is given as [18] dn w c N w (1 c ) N w I D dt ev e c s (.16) dc c 1 Nw (1 c ) c c a( c ) E ( t ) (.17) dt D e c s CH c is the occupation probability of ground state, N w 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 15

33 Chapter Two Theory of CH in QD SOA injected current, E(t) is the electric field of the interacting light, a is the absorption renormalized for the occupation probability, according density matrix theory an is given as [18] i a( ) (.18) c vc, i cv, i cv, i vc, i V i According density matrix theory, the transition between conduction band (CB) and valence band (VB) is govern by the following equation [18]: d cv dt 1 i ( i ) cv ( 1) (, ) (.19) cv c v E z t cv cv is decoherent time, i is transition frequency, the transition energy is given by [3] i Ecd, i E E g (.) vd, i The band gap-shrinkage effects (the dependence of n Eg 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, v c c and denote the electron and hole occupation probabilities in the conduction and valence bands, respectively. Fig..: Semiconductor band structure [3]. 16

34 Chapter Two Theory of CH in QD SOA.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 and a weaker probe at 1, the fields can be mixed nonlinearly to produce a so-called conjugate signal at ( 1 ) as in Fig.3. consider a total electric field propagating in the SOA of the form [3] E z t E z e E z e E z e c c o 1 (, ) ( ) i t ( ) i t ( ) i t. (.1) 1 E is the electrical intensity of the pump, E 1 is that of the probe and E is the conjugate formed through nonlinear mixing [3]. Optical Field 1 Fig..3: Optical field of pump, probe and conjugate versus frequency [3]. The field E z, tinduces a polarization, amplifier [3] P z t P z e P z e P z e c c P z t in the active region of the o 1 (, ) ( ) i t ( ) i t ( ) i t. (.) 1 The relation that relate between the polarization and dipole terms is given by [18] 1 V,.3 P z t j i, k cv, j cv vc The dipole terms take the form [3] i 1,, o t i t i t e,1 e, e (.4) cv j j j j 17

35 Chapter Two Theory of CH in QD SOA 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] it * it c, j c, j c, j e c, j e (.5) N N N e N e it * it w, j w, j w, j w, j (.6) The variables on the right hand side of the Eqs. (.5) and (.6) are time independent, the solution of Eq. (.4) is achieve by substituting Eqs. (.5 and.6) in Eq. (.19), one obtain [3] cv, j * * it cv, j ( ) j o ( c, j v, j 1) Eo ( c, j v, j ) E ( c, j v, j ) E 1 e cv, j i1t i ( 1 ) ( c, j v, j 1) E1 ( c, j v, j ) E e cv, j * * i t i c, j vd, i c, j v, j ( ) ( 1) E ( ) E e (.7) where j ( ) is the Lorentzian lineshape determined by the decoherence time and is responsible for homogeneous broadening [3] 1 j ( ).8 ( ) i j The dipole terms, Eq. (.4), can be extracted by the comparison between Eqs. (.7) and (.4), to get [3] (, c, j v, j 1) Eo ( c, j v, j ) E cv j cv, j ( o ).9 * * ( c, j v, j ) E1 cv, j cv,1 j ( 1 ) ( c, j v, j 1) E1 ( c, j v, j ) E (.3) cv, j * * cv, j ( ) ( c, j v, j 1) E ( c, j v, j ) E (.31) ( c, j v, j ( 1) 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. (.17) for conduction and valance band, one obtains 18

36 Chapter Two Theory of CH in QD SOA N w in 1 D c 1 (.3) i in * 1 ( cv j ) j ( ) E c, j v, j Where in 1 N wv 1 1 D c e CH 1 (.33) 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 F F Eq. (.3), one obtains N D,, 1 1 [18]. By taking E in c j v j c v w F F c v in c The derivation of eq. (.34) gives d c, j d v, j in 1 in 1 N w.35 dn w dn w D c D c By comparing the result in Eq. (.33) with the result in reference [18] for QD SOA, here, the above equation is differ from ( Eq. (8) in [18]) by the term of carrier heating time constant ( CH burning time constant is equivalent ). In QD structure, the time of spectral-hole N wv 1 Dc e 1 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.(.17) can be written as [18, 3] 19

37 Chapter Two Theory of CH in QD SOA d c dt L f 1 N (1 f ) f c f c c c w c c c c a( c ) E ( t ) D e c s CH (.36) Where f c is the Fermi function which is a function of temperature f t F( T t ) (.37) c is Fermi function at lattice temperature, the steady state and small signal of Fermi function can be written as [3] f f f T i t ck, Fc c, k ( c exp( )) (.38) Tc 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.(.5,.6 and.38) in Eq. (.36) and combine for conduction and valence bands, one obtain 1 N 1 1 w N w 1 1 N w 1 1 i c, k v, k ( ) c, k v, k ( ) e D c CH D c e D c CH f f N 1 N 1 1 f T f T w w c, k v, k c, k v, k ( c v ) D c e D c CH Tc Tv i k i ħ * * * * cv,, 1 χˆ ω1 χˆ E1 χˆ ω χˆ c k v k E E E ) ρc,k ρ v,k * χˆ χ ˆ ω E 1 w,,,, (.39) w c, k v, k c k v k { N w [( ) c k v k 1 ( )] ( Tc Tv ) (1 i in ) D c D c Tc Tv 1 N 1 N 1 iin k * * * *,, 1 χˆ ω1 χˆ E1 χˆ ω χˆ c k v k E EE ) f (.4) f

38 Chapter Two Theory of CH in QD SOA The small signal of Eq. (.16 and.4) and using Eqs.(.5,.6 and.38), the result i 1 i N w in * * * * N 1 χˆ ω1 χˆ E1 χˆ ω χˆ w X E E E )* D V i D c i * E χˆi ω1 ˆχ iin k * {1 E χ ˆ ω1 χ ˆ i i } i 1 i * in N w in [(WY XZ) X E χˆ ω1 χ ˆ i i [1 ] D V i D c D c i 1 X DV i f c, k f v, k i ( Tc Tv ) Tc Tv i * in N w in i 1 i i c c i 1 [(WY XZ) X E χˆ ω χ ˆ [1 ] D V D D (.41) Where 1 1 N w 1 1 Y i e D c s CH (.4) 1 1 c Z (.43) D c W 1 1 c 1 ( i ) (.44) e s c CH The linear and nonlinear polarization (P, P 1 and P ) is simply estimated by substitute of the dipole terms in Eq. (.3), then, comparing it with Eq. (.). Separating terms of different resonances, the orders of polarization are given by [3,18] P 1 cv, j j o c, j v, j E V j ( )( 1).45 1 P ( ) ( 1) E ( ) E (.46) cv, j 1 j 1 c, j v, j 1 c, j v, j V j 1 P ( ) ( 1) E ( ) E (.47) cv, j * * j c, j v, j c, j v, j V j 1

39 Chapter Two Theory of CH in QD SOA 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 (.3) written as N w in 1 1 D P ( ) χ ω ( ) (.48) j c t ˆ j E V j i, k iin k * { 1 E ˆ χ ω χ ˆ i i } The linear susceptibility is simply extracted by comparison with P 1 E X ( l ) N w in i D c χˆ ω ( j ) (.49) V j i, k iin k * { 1 E ˆ χ ω χ ˆ i i } Substituting Eqs.(.4) and (.41) in Eq.(.46), the second-order polarization is given by * χˆ P ( t) ω E χ ω { [( ) 1 i in ˆ j Nw c k v k (l) ,, V j i, k (1 i in ) D c 1 1 i 1 f c, k f in v, k ( )]} χˆ j ω 1 {( Tc Tv)} D c V j i, k (1 i in ) Tc Tv j i, k 1 1 i χˆ ω1 c, k v, k 1 V ( χ ˆ ω i in k j ( 1 iin) 1 * * * E1E χ ˆ ω χˆ E E )E (.5) The induced polarization density in Eq.(.5) 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 P is identical to that of P 1. For simplifying the density polarization is expressed as [18,3,34]

40 Chapter Two Theory of CH in QD SOA E P X E X E X E E CDP CDP * 1 L 1 1 1; ; 1 1 1; ; SHB SHB * X 1; ; 1E1 X 1; ; E CH CH E * X 1; ; 1 E1 X 1; ; E (.51) E E P X E X E X E E CDP CDP * L ; ; ; 1; 1 SHB SHB * X ; ; E X ; 1; E 1 CH X E X 1 E 1 CH * ; ; ; ;.5 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 g i 1 N c n V j D c j w * in 1 j ( ) j ( ).53 dg 1 j in 1 in ( i) 1 Nw j( ).54 dn c n V j D c D c Eqs.(.53) and (.54) have been derived by substituting the relation of c, j v, j 1 and d dn c, j v, j w d dn w E E E E in QD system identified into Eqs. (.34,.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 (.49). Substitute Eqs. (.41, ) into Eq. (.5) and then compare the result with Eqs.(.51-.5), we can determine generalized susceptibilities due to CDP and SHB as 3

41 Chapter Two Theory of CH in QD SOA X X CDP SHB ; ; q m n ; ; q n m cn dg ( q ) s g( ) E q dn q i 1 i nm in D s E WY XZ l X E sat i E in V 1 i 4 mn in N 3 (.55) w * j ( q ) in 1 j ( m ) j ( n ) (.56) D c l E is the saturated field for two-level QDs system which can be defined as [18] sat E l sat dg cn s dn l w.57 The susceptibility due CH is derive as X CH 1 i 1 f c, k f v, k χˆ q m n j ω q { Tc Tv } V j i, k (1 i inmn ) Tc Tv (,, ) (.58) To calculate the temperature at small signal, we must use the definition of energy density (U) which is given by the following equation [3] U x t x, j x, j t (.59) j Multiplying (.) by x, j, and summing over j, one obtain Nw ( E x U ) 1 x e D c s CH V U U U i U K E ( z, t ) Ex( vc,icdvd,i cv,icdvd,i ) E(t) i (.6) The term ( K x E ( z, t ) ) is phenomenologically added to represent the contribution of CH induced by FCA, is a coefficient that can be express by 4

42 Chapter Two the cross section Theory of CH in QD SOA for free carrier absorption (FCA) in the conduction and valence bands which is given by [18] K x n ng g x N w (.61) Where x is the cross-section, is the permittivity of free space, n is the refractive index, ng 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 [3]. i t U U h T e c. c (.6) x x x Substituting Eq. (.6) and Eq.(.1)in Eq.(.6), to obtain hx T x exp i t 1 N w i hx T x exp i t hx T x expi t D h T exp i t h T exp i t 1 i x x x x cv K x ( E1E E E ) exp i t s CH V 1 i V cv * * * * E ˆ ˆ ˆ ˆ x ( c, k v, k 1).{ χ ω1 χ E1E χ ω χ E E }exp i t * E x (,, χˆ ω χˆ c k ) v k E 1 exp i t e c (. 63) For simplicity, we neglect the last term in Eq.(.63) and used Eq.(.18), then one gets 1 iin cv Tx hx i in 1 ink x ( E1E E E ) E x ( c, k v, k 1). V * * * * { χ ˆ ω1 χˆ E1E χ ˆ ω χˆ E E} (.64) 1 inhx 1 i cv x i in V 1 ˆ * * * T { (,, ) χˆ x E c k v k 1 ω χ (E1E E E ) 1 K ( E E E E )} (.65) x 5

43 Chapter Two Theory of CH in QD SOA From definition of the material gain (Eq.(.53)) and free carrier absorption factor, Eq.(.61) can simplified as h c T { E g( ) N }( E E E E ) (.66) 1 in x x x g g X 1 1 i in Substituting Eq.(.66) in Eq.(.58) and use the identity ω c g ω cv f x, k ˆ k x x x 1 χ ω ( ) ( T (ω) i ) (.67) x V T T T The susceptibility due to CH will become 1 ( (ω) ) CH c g ω T i x inh x g g T x 1 i in 1 i in g( ) XN w [ E x ]( E1E E E ) (.68) g( ) 6

44 Chapter Three Four Wave Mixing and Pulse propagation

45 Chapter Three FWM and Pulse Propagation 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 [5]. One of these ways which can be used is four-wave mixing (FWM). It is a promising technique that can replace multiwave converters by a single one [51]. It is typically realized in semiconductor optical amplifiers (SOAs) and requiring an external pumping sources [5]. 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 [6]. 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 7

46 Chapter Three FWM and Pulse Propagation well-known phonon bottleneck effect [6]-[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,,3,3,67], 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.. 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 8

47 Chapter Three FWM and Pulse Propagation 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 phononscattering, 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 9

48 Chapter Three FWM and Pulse Propagation 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 [7], 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 E E (3.1) c t c is the velocity of light, is the dielectric constant which is given by [71] n X ( N) 3. b n b 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 [7] n X ( N ) N i g ( N ) (3.3) p is frequency of the emitted photon, n is the effective mode index, gn ( ) is the optical gain approximately varies as [71] g ( N ) dg N N (3.4) dn where dg dn is the differential gain, Γ is the optical confinement factor, N is the injected carrier density and N 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] 3

49 Chapter Three FWM and Pulse Propagation 1 ˆ i ( k z t ) E x y z t F x y A z t e c c (,,, ) (, ) (, ). (3.5) where ˆ is the unit vector of polarization, F x, y is the waveguide-mode n distribution, k, and A(z, t) is the slowly varying amplitude of the c propagating wave. Substitute Eqs.(3.5-3.), 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] d F d F dx dy c o n (3.6) b n F da 1 da io 1 XA dz dt nc g int A (3.7) int is the loss coefficient, g is the group velocity ( c / n ) g g, while n g is the group refractive index. In bulk, the time evolution of carrier density (N) describes by the following equation [71] dn I N g ( N ) A dt ev c m (3.8) 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] dg g g g A (3.9) dt E c sat g is the small signal gain, which is given by [71]: dg I dg I go N o 1 N o dn evn o dn I o c where I is the current required for transparency. I o evn o c

50 Chapter Three FWM and Pulse Propagation E sat is the saturation Energy, Eq. (3.7) can be further simplified by using the retarded time frame[71] z t g i t Then, assume that A Pe and using Eq. (3.4), one obtain [71] (3.1) da g (1 i) A 3.13 dz dg g g gp d E 3.14 c Sat dp 1 ( g ) int P dz d 1 g dz 3.16 where P(z,τ) and φ(z, τ) are the instantaneous power and the phase of the propagating pulse, respectively. The quantity Esat Psat c. Psat power of the amplifier which is given by [71]. P if sat m a int c is the saturation 3.17 The solution of Eqs. ( ) generally requires some approximations, g, it is possible to solve these equations in a closed form. In the following, assume int =. Eqs.( ) are then integrated over the amplifier length to give [71]. P P e h ( ) out ( ) in ( ) out ( ) in ( ) h( ) 3.19 where P and are the power and phase of the input pulse. The function in h is defined by [71] L in (, ) 3. h g z dz 3

51 Chapter Three FWM and Pulse Propagation 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] dh() g L h( ) Pin 3. d E c sat h ( ) e 1 1 Numerically, For a given the gain ( g ) and input pulse shape Pin (as an example, consider a Gaussian pulse), Eq.(3.1) can be solved to obtain the gain integral. The output pulse shape is then obtained from Eq.(3.18). Also, Eq.(3.1) 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. (3.1) 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 (.-.3 ns), and the above approximation is valid for p < 5 ps. In the p limit 1, the solution of Eq. (3.1) is [71] c U in ( ) 1 Esat h( ) ln 1 1 e 3. G o where G exp( gl) is the unsaturated single-pass amplifier gain and Uin() represents the pulse energy which is given by the following equation[71] U in ( ) Pin ( ) d 3.3 By definition, U ( ) in E pulse the solution of Eq. (3.3) is given by [71] U in in, where Ein is the input pulse energy and for Gaussian E in ( ) 1 erf (3.4) where erf is the error function and τ is pulse width at half maximum. 33

52 Chapter Three FWM and Pulse Propagation 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 -level rate equations [18]. From Eq. (.16) and using Eqs.(3.4, 3.1 and 3.11) the time evaluation of gain is derive as [] dg go g QD g 1 (3.5) d c gmax where QD SHB D SHB dg dn 1 N w e D c 1 g max is the maximum value of gain. (3.6) (3.7) 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 34

53 Chapter Three FWM and Pulse Propagation relaxation rate [18]. The integration of Eq.(3.5) over the amplifier length (L), one obtain [] d h( ) gol h() QD h( ) L d c gmax (3.8) 3.5. FWM pulses If we have two injected pulse signals (assuming transform-limited Gaussian pulses) represented by [13] E E 1 A exp 3.9 A exp where E is the pump signal at, E1 is the probe signal at 1, and is the delay time between the two pulses. The FWHM of pump and probe pulses are and 1, respectively. The optical field at the input facet (z = ) is [13] i t, 3.31 E z t E E e 1 Note that is the phase between E and E 1. Integrating Eq. (3.31) yields [13] h 1 i E ( L, ) E (, ) e 3.3 out in The small signal analysis can be applied to Eqs. (3.31) and (3.3), this treatment leads to [13] h 1 i 1 E ( L) e 1 F ( ) E E 3.33 h 1 i 1 1 E ( L) e 1 F ( ) E E 3.34 h 1 i * 1 E ( L) e F ( ) E E

54 Chapter Three FWM and Pulse Propagation F C e h 1 1i 1 ix x E x SHB, CH 1 i x 1 i E sat 3.36 In the expressions above, the field intensity, amplifier saturation intensity E sat E, is normalized to the, C is the phenomenological parameter to compensate for the nonplanar nature of the waveguide [13]. Eq.(3.35) describes the wave mixing product, E, whose frequency is, and is the gain recovery time. Other products are also created but are all much smaller than E 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 (CH c ), of holes in the valence band (CH v ), and spectral hole burning (SHB). Each process has a characteristic time constant, corresponding nonlinear gain coefficient x, a 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] F e 1 1 i 1 i h QD N x x C E 1 x i x E sat where

55 Chapter Three FWM and Pulse Propagation 1 D N w N w 1 D N w 1 1 D s i i e c c s e Dc s e c D c (3.38) 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 (1 i ) X X SHB SHB CDP SHB Normalize 4 cv N w * 3 j ( ) in 1 j ( n ) j ( m ) in j D c dg ( ) cv N * cn 1r w dn 1 in ( ) ( ) D c 1 w, x in x x w CH, x ich E K T 1 i in s g (3.39) N E h N (1 ) (3.4) 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. * me is as the effective mass for the electrons or holes * k T m and is the effective height of the quantum-dot layer. h x ( x 3 ) is the heat capacity of the free electrons assuming a two-dimensional (D) electrongas model [18]. 37

56 Chapter Three FWM and Pulse Propagation 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 (1Gb/s) [8]. 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] eff E h e F L E sat,

57 Chapter Four The Results and Discussion

58 Chapter Four The Theoretical Results 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 [8]. 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 [18,8,3,37]. This thesis not just about theory of CH without the other mechanisms, it gives us the interaction between them. 4.. 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 In.53Ga.47As which was lattice matched to GaAs * * and operating around 1.3 μm. The material parameters are m e =.3 m, m v =.41 m, g =.345 ev, c 1 ps, s. ns, cv 15 fs, c = m -, v, the amplifier length (L=3 μm), width (w d = μm), the thickness of each layer is (L w =1 nm),.7 and I=5 ma. These values are similar to those used in [18,3], in most of the results to be presented, we consider a pump wavelength of 1.33 μm, corresponding to a photon energy, =.93 ev, which is at the peak of the gain curve. 39

59 Chapter Four The Theoretical Results 4.3. Occupation probability of dot level The occupation probability of Gs has been calculated by using Eq. (.3), 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 with d in the model presented by [18], in previous model time of SHB which is very fast compared to in. d CH compare represents the Fig. (4.1): The GS occupation probability versus carrier density 4

60 Chapter Four The Theoretical Results 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. (.53). The effect of CH is present in our calculation, this is obviously shown in Fig. (4.). 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. ): The gain versus wavelength 41

61 Chapter Four The Theoretical Results 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. 4

62 Chapter Four The Theoretical Results 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.(.16 and.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. 43

63 Chapter Four The Theoretical Results Fig.(4.5): The time domain of carrier density Fig. (4.6): The occupation probability versus time 44

64 Chapter Four The Theoretical Results 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: 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. 45

65 Chapter Four The Theoretical Results 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 NWL. CH is increase with increasing carrier density and reduced at high detuning. at low detuning ( (1-1) GHz) the dependence of detuning is very weak, with increasing of detuning (> 1 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 considered a calibration of intraband relaxation time. in which can be In Fig.(4.1),a comparison has been done between our model (QD) and the bulk model in [18]. At low carrier density, the CH effect is completely 3 match, but with increasing of carrier density above the value ( N 1 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. w 46

66 Chapter Four The Theoretical Results Fig. (4.8): 3-dimenssional plot of CH NWL. Fig.( 4.9): The effect of CH on CH NWL curves 47