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Modeling of Quantum Dot Semiconductor Optical Amplifiers for Telecommunication Networks vorgelegt von Diplom-Ingenieur Dmitriy Puris von der Fakult¨ at IV - Elektrotechnik und Informatik der Technischen Universit¨at Berlin zur Erlangung des akademischen Grades Doktor der Ingenieurwissenschaften Dr.-Ing. genehmigte Dissertation Promotionsausschuss: Vorsitzender: Prof. Dr. Lang Gutachter: Prof. Dr. Petermann Gutachter: Prof. Dr. Leuthold Tag der wissenschaftlichen Aussprache: 31.01.2013 Berlin 2013 D 83

Modeling of Quantum Dot Semiconductor Optical … of Quantum Dot Semiconductor Optical Ampli ers for Telecommunication Networks vorgelegt von Diplom-Ingenieur Dmitriy Puris von der

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Page 1: Modeling of Quantum Dot Semiconductor Optical … of Quantum Dot Semiconductor Optical Ampli ers for Telecommunication Networks vorgelegt von Diplom-Ingenieur Dmitriy Puris von der

Modeling of Quantum Dot Semiconductor Optical Amplifiersfor Telecommunication Networks

vorgelegt vonDiplom-IngenieurDmitriy Puris

von der Fakultat IV - Elektrotechnik und Informatikder Technischen Universitat Berlin

zur Erlangung des akademischen Grades

Doktor der IngenieurwissenschaftenDr.-Ing.

genehmigte Dissertation

Promotionsausschuss:Vorsitzender: Prof. Dr. LangGutachter: Prof. Dr. PetermannGutachter: Prof. Dr. Leuthold

Tag der wissenschaftlichen Aussprache: 31.01.2013

Berlin 2013D 83

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Contents

1 Introduction 51.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Quantum dots 72.1 Growth technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Band structure and energy levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Quantum dot semiconductor optical amplifier . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 Carrier dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Time-domain modeling 173.1 Time-domain modeling fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Conventional time-domain models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3 Propagation Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.4 Internal losses, two-photon absorption,

Kerr effect and free-carrier absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.5 Diagram of the presented time-domain modeling . . . . . . . . . . . . . . . . . . . . . . . 23

4 Digital Filters 254.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.2 Filter equation in Z space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.3 Structures of gain filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5 Modeling of QD-SOA static behavior 335.1 Amplified spontaneous emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.1.1 Carriers dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.2 Continuous wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

6 Modeling of QD-SOA dynamic behavior 396.1 Pump-probe behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

6.1.1 Cross-Gain Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396.1.2 Cross-Phase Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

6.2 Frequency chirp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456.3 Four-wave mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

6.3.1 Four-wave mixing between two continuous waves . . . . . . . . . . . . . . . . . . . 486.3.2 Four-wave mixing between two pulse sequences . . . . . . . . . . . . . . . . . . . . 50

6.4 Propagation of ultrashort pulses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526.5 Improvement of linear amplification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

7 Conclusions 577.1 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

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A Calculation of intraband transition rates 61

B Acronyms 63

C Coefficients 65

D List of main symbols 67

Bibliography 69

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Chapter 1

Introduction

Optical amplifiers are one of the key parts of the telecommunication systems. Their active mediumincreases number of the photons due to the stimulated recombination and thus increases the opticalpower. New created photons have the same direction and the phase as the initial ones, so an amplificationkeeps the information, carried by the photons.

There are two main applications of the amplifiers - the linear amplification, when the device linearlyincreases the optical power, and the signal processing, when the device changes one signal correspondingto the power or the phase of another one. The linear amplification is used mainly to send a signal overlong spans with significant losses. The signal processing is used to change a wavelength of the signal, toregenerate signal to its initial shape and suppress distortions, in all-optical logic elements and for otheraims.

The semiconductor optical amplifiers (SOAs) are one of the most important parts of the opticalnetworks. SOAs are based on the active semiconductor medium, similar to semiconductor lasers. Theirmain advantages are low energy consumption and small size, which makes it possible to integrate theminto large circuits.

There are two main conventional types of the semiconductor optical amplifiers: bulk and quantumwell (QW). The sizes of their active regions are significantly larger rather then electron wavefunctionsthat creates continuous energy bands and broadband gain spectrum.

In the past decades, semiconductor quantum dots (QD) were deeply investigated [1–4]. Quantumdots are semiconductor heterostructures with a size of tens of nanometers, the same order as the carrierwave-functions that produces discrete energy levels for carriers in the active region. In the quantum-dotsemiconductor optical amplifiers (QD-SOA) the material gain is produced by carrier-photon interactionon these levels that may provide multiple benefits as compared to the conventional SOAs, like highsaturation power, fast material gain recovery, chirp-free amplification etc.

Most of the created QD-SOAs have the main transition wavelength around 1.3 µm, which makesit possible to use them in the corresponding telecommunication systems. Low dispersion on thesewavelengths makes it possible to transfer broadband high-speed signals without dispersion compensators.

Creating a QD-SOA for an operating wavelength 1.55 µm is problematic as by using the same materialas a 1.55 µm bulk SOA the QD-SOAs will have shorter transition wavelength due to discrete energystructure. Development of this device is still possible by using another material or changing materialstrain and QDs size. The presented QD-SOA model is phenomenological and can be used to operate with1.55 µm QD-SOA as all physical processes will be the same and only a simple correction of coefficientswill be required.

1.1 Motivation

To model a QD-SOA behavior it is required to have a general model that will include all of importanteffects and is appropriate to work with different types of the optical signals. As in many publications

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QD-SOAs are described only partially [5, 5–24], it is required to collect all presented knowledge andsynergetically connect them. The presented work is devoted to the description of this general model.The general model is required to determine the influences of every included effect, and also to find thepossibilities of QD-SOAs utilization in optical networks.

As carrier dynamics in QD-SOAs were well described, there is still a lack of an appropriate model forthe optical signal propagation. Phase-based effects were not described widely enough in most publications,so they are one of the main aims of the presented work.

The main requirements for the model are as follows:

• The model should operate with numerical simulations. This will increase calculation time comparedto analytical computations, but will significantly simplify usable equations and will allow to add allrequired physical effects.

• The model should operate with a signal in the time domain as the aim of the calculation is toreceive values of the optical power and phase over time for the output signals.

• The model should have modular structure and the influence of each effect for a single time momentshould be calculated separately from other effects, based only on the values of the carrier distributionsand the optical signal at this time moment. This will allow to calculate influence of any single effectas well as combined actions of all effects in the semiconductor medium.

• The model should include frequency dependence of the material gain and corresponding phasemodulation.

As this model will be verified by various types of experiments, it is not possible to change all coefficientsfreely, as is in other works. The same coefficients are used for every simulation, so improving one of themwill destroy all others. This makes it not possible to receive a perfect agreement with the experimentsfor all simulations, but we have determinated the importance of each parameter and process and its rolein the whole signal modulation, as it will be described below.

This thesis is organized as follows. In chapter I, the basic theory of QDs and carrier dynamics arebriefly described. In chapter II, the basic ideas of the time domain modeling are given as well as acomparison of the presented model with conventional ones and the description of the operation principlesof the presenting model is provided. In chapter III, the fundamentals of digital filters and their utilizationin the model are described. In chapters IV and V static and dynamic experiments and simulations arepresented. Conclusion is provided in chapter VI.

Appendix A provide an approximation to calculate carrier transition rates. Appendix B - list ofacronyms. Appendix C - list of coefficients used in the model. Appendix D - list of symbols in thepresented thesis.

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Chapter 2

Quantum dots

Described quantum dots (QDs) are semiconductor crystals that have size in a scale of 10 nm [1,2,25–31].Figure 2.1 demonstrates three-dimensional scanning tunneling microscopic images of a single uncoveredInAs QD grown on GaAs. QDs are large enough to hold electrons and holes, and small enough to havea significant separation between energy levels. Compared to bulk and QW structures with continuousenergy bands, QDs emit and absorb photons only with discrete frequencies, as presented in Fig.2.2.

Figure 2.1: Three-dimensional scanning tunneling microscopic images of an uncovered InAs QD grownon GaAs (a) and image in 110 crystal direction (b) [27].

The discrete nature of energy levels provides many benefits, like stable gain peak frequency for differentcarrier densities and active region temperatures. The optical frequency of the highest amplificationdepends on the inhomogeneous broadening of QDs, while in structures with continuous energy bands itdepends on the carrier distributions, as schematically presented in Fig. 2.3.

Another benefit is the behaviour of the phase modulation. As the material gain from a single QDenergy level is perfectly symmetric around the peak gain frequency for homogeneous and inhomogeneousdistributions of QDs, the differential refractive index is exactly zero at this energy as presented in Fig. 2.4.

2.1 Growth technique

The main technique to create a quantum dot structure for SOAs and lasers is the Stranski-Krastanowgrowth. In this technique In and As atoms are self-organized into nanoscale islands to minimize theirpotential energy. The growth of these three-dimensional structures occurs due to the minimization of thestrain and surface energies of the system [32]. This technique allows to simplify the growth process ascompared to the lithography.

Figure 2.5 schematically presents a conventional QD layer between GaAs barriers and the dot-in-a-well(DWELL) heterostructure. In the DWELL structure a QD layer is covered by an additional thin InGaAs

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ρ

E

3D 2D 1D 0D

3D2D1D0D

Figure 2.2: Schematic comparison of 3D bulk structure, 2D quantum well, 1D quantum wire and 0Dquantum dot. Structures are presented above and corresponding densities of the states are presentedbelow.

f

f f

f

g

gg

g

T1T2

T1<T2N1N2

N1>N2

N1N2

N1>N2 T1T2

T1<T2

(b)(a)

Figure 2.3: Schematic comparison of bulk (above) QD- (below) SOAs gain profile for different carriersdensities (a) and carriers temperature (b). Changing of carrier densities or temperature in bulk SOAshift peak gain frequency, but does not shift in QD-SOA.

capping that is working like a quantum well. The DWELL structure changes strain in QDs and therebydecreases transition energy of the GS level. This makes it more appropriate for telecommunication aims.

2.2 Band structure and energy levels

For a bulk semiconductor structure we can assume the effective-mass approximation, where the detailedconduction and valence band structures are approximated by a simple parabola. The energy levels forelectrons in the conduction band Ee and for holes in the valence band Eh have a dependence on thecarrier momentum (wave vector) k [33]:

Ee =~2k2

2me+ Eg0 (2.1)

Eh =~2k2

2mh

where me and mh are the effective masses of the electron and hole respectively and Eg0 is the bandgapenergy.

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Figure 2.4: Calculated differential gain g′ and differential refractive index n′real for quasi-ideal QDensemble with a symmetric gain curve and only one energy for electrons and holes [25].

GaAs

GaAs

InAsInInGaAs

Barrier

BarrierGaAs

GaAs

InAs

Barrier

Barrier

Quantum DotsWetting Layer Wetting Layer

(a) (b)

Quantum WellQuantum Dots

Figure 2.5: Schema of a conventional QD structure (a) and DWELL where QD layer is covered by anadditional quantum well (b).

Wave vector k has a discrete nature and is determinated by the size of the confinement structure:

kx =2πn

Lx(2.2)

where n is an integer ranging from −∞ to ∞ and Lx is the length of the semiconductor crystal in thex direction. The total wave vector is:

k2 = k2x + k2

y + k2z (2.3)

where kx,y,z are wave vectors for x, y and z directions.As the size Lx of the structure is significantly larger than the electron wavefunction, we can assume

a continuous distribution of energy levels for a 3D electron gas and corresponding density of states in abulk semiconductor. This gives the number of states ρ between Ee(h) and Ee(h) + dE calculated as [33]:

ρ(Ee(h)) =1

(2me(h)

~2

) 32 √

Ee(h) (2.4)

In the QDs the electron wave function is limited in all spatial directions, which increases energyseparation between levels. The distribution of the energy levels in QDs is determined by multipleparameters like the materials of QDs and the surrounding area, a size and a shape of QDs, materialstrain, carrier densities etc [1,2,28,29,31,34–36]. Examples of the calculated energy structures of a singleQD are presented in Fig. 2.6. As it is not an ordinary task we will not consider it in the present work.

In the present work we will consider only two main energy levels of QDs - ground state (GS) and thefirst excited state (ES), because only they were visible in the experiments. Values of the energy levels inthe present model are shown in Fig. 2.7a.

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or only on the biexciton ground state. Such dots must besufficiently small so that confinement effects are dominant,electron levels are widely separated, excitons are weaklycorrelated,58 and Coulomb interaction can be treated as acorrection of the kinetic quantization effects.65 Therefore werestrict ourselves to calculate the exciton and two-electronground states within the Hartree approximation, where ex-change and correlation interactions are discarded. Correc-tions for biexciton states are of the order of 2 meV.66 Theelectron term scheme governs the main structure of the re-combination spectra while the less widely separated hole lev-els contribute to the fine structure.

We model the linear absorption spectra of single QDs,which to a certain degree can be measured by photolumines-cence excitation,67 calorimetric absorption,1 and transmissionspectroscopy.68 Within the dipole approximation the linearabsorption coefficient of the QD is proportional to27,53

I ab5pe2\2

e0m02

2

Vue•pabu25

2pe2

e0V UeK aU]H

]k UbL U2

, ~18!

wheree is the direction unit vector of the electric field of thelinearly polarized incident light,V is the QD volume,H isthe QD Hamiltonian from Eq.~12!, k is the wave-numberoperator,pab is the momentum matrix element of the transi-tion from stateua& into stateub&, e is the electron charge,m0 the free electron mass, ande0 the vacuum permittivityconstant. Inserting the envelope functions into Eq.~18! isequivalent to considering the central cell part ofpab onlywhile neglecting the envelope part,23,53a commonly acceptedmeasure. We note that Eq.~18! provides the full treatment ofthe structural,k-, and strain dependence ofp, i.e., the infor-mation on the QD is not only stored inua& and ub&.

Since spin-orbit coupling is neglected, and due to the ab-sence of magnetic fields, all energy levels in the QD are spindegenerate: IfC1 according to Eq.~13! is an eigenvector toEq. ~12! then

C25S 04 214

14 04D C1, ^C1uC2&50 ~19!

is also an eigenvector (04 and 14 are the 434 zero andidentity matrix, respectively, andC1 is the complex conju-gate ofC1). This is not the Kramers symmetry and also truefor BÞ0 ~Kane parameter! in Table II. The momentum ma-trix elements for nonexcitonic transitions are obtained byincoherent averaging over the degenerate eigenspacesa andb, giving

ue•pabu25 14 ~ ue•pa1b1

u21ue•pa2b1u21ue•pa1b2

u2

1ue•pa2b2u2!, ~20!

whereua1& andua2& (ub1& andub2&) satisfy Eq.~19!. On thelevel of simplification persued here, the exciton recombina-tion is modeled the same way, just using self-consistentstatesua& and ub& according to Eq.~14! instead. At self-consistency the eigenspacesa andb are not orthogonal anymore since they belong to two Hamiltonians differing in theirconfinement potential parts. However, forI ab this does notmatter since]H/]k is not altered by changing electrostatic

potentials. The oscillator strength obtained this way is alower bound for the true excitonic oscillator strength, sincethe actual two-particle character of the exciton69 was lost bymaking the ansatz of a separable exciton wave function.However, we expect correct modeling of the reduction of theoptical anisotropy due to exciton formation.

A. Electron-hole transitions

Figure 9 shows the linear absorption spectra for the inter-action of QDs witheuu@100# polarized light, dependent onthe dot size and the strain model. The dimensionless oscilla-tor strength

I 52

m0EpInAs

ue•pabu2 ~21!

refers to the bulk optical matrix parameterEp of the QDmaterial InAs and is the eight-bandk•p analogon to theoverlap integral caucb& in the effective mass approxima-tion. All corrections due to excitonic effects are neglected inthis plot, and the transitions have been broadened artificiallyby Gaussians with 10 meV FWHM. Since at least for the

FIG. 9. Linear absorption spectra~electron-hole transitions, noexcitonic correction! of InAs pyramid QDs of different sizesb forlinearly @100# polarized light, calculated for strain distributions ac-cording to the CM~black! and VFF~gray! models. The arrows with‘‘quantum numbers’’ refer to the black curve~CM! and indicate thestrongest contributing transitions. The oscillator strengthI is givenby Eq. ~21!. The absorption lines are artificial Gaussians with 10meV FWHM. The seemingly missing transitions forb520.4 nm atenergies above 1200 meV have not been calculated.

PRB 59 5697ELECTRONIC AND OPTICAL PROPERTIES OF . . .

JIANG AND SINGH: SELF-ASSEMBLED SEMICONDUCTOR STRUCTURES 1193

(a)

(b)

Fig. 2. (a) Electronic spectrum for an InAs–GaAs dot with base diameter 80A and height 42A calculated by the simple effective mass approach [2]. Solidlines represents states that can be observed in photoluminescence spectra anddashed lines represents states that cannot be observed in photoluminescencespectra. (b) Electronic spectrum calculated for the dot with base width 124A and height 62A using the effective mass approach for conduction band(m� = 0:04m0) and a four-bandkkk � ppp model for valence band [7].

layer states in this state because it is less confined. Thedirection of excitation is in the direction.

As found by [7], we also find many confined hole states.The splitting between the ground and excited states are from22 to 30 meV. There are no wavefunction overlap betweensecond excited hole states and second excited electron statesdue to different excitation directions.

In Fig. 4 we show our results for the Stark effect. The solidline is for the case where the field is along the [100] direction

Fig. 3. Electronic spectrum for a InAs–GaAs dot with base width 124A andheight 62A calculated using the eight-bandkkk �ppp model described in the text.

and the dotted line is for the case where the field is alongthe [001] (growth) direction. In Fig. 4(a) we show how thetransition between ground states shifts with field. We see thatwhen the field is in the growth plane the Stark effect is quitestrong. There is a red shift of about 22 meV for a 100-kV/cmfield. When the field is along the [100] axis the Stark effectis symmetric with respect to the orientation of the field due tothe symmetry of the dot.

For the case of the field parallel to the growth direction,the Stark effect is relatively weak. Also, the effect can causeeither blue shift or red shiftdepending on the field polarity.When the field is in the growth direction, there is a blue shiftwhen the field is not too large. This occurs because the electricfield pushes the heavy holes to the top of the dot where theyhave less confinement [Fig. 1(b)]. When the field is greaterthan 150 kV/cm, the effective bandgap begins to decrease.When the electric field is in a direction opposite to the growthdirection, there is the usual red shift.

In Fig. 4(b) we show the effect of the electric field onthe intersubband separation in the conduction band of theInAs–GaAs dot. We see that for the case when the field is alongthe [100] direction there is an decrease in the intersubbandseparation. The decrease is not very large but can be used formodulating intersubband absorption. When the field is alongthe growth direction, we again see an asymmetric (with regardto the field orientation) shift.

In Fig. 5 we show the scattering rates involving two LOphonons for the ground and first excited states. We see thatwhen the energy separation between the first excited andground states is equal to the the scattering timeis 0.1 ps. However, the rate drops off very rapidly when

are totally determined by Jss and the single-particle energyspacing �.

However, real self-assembled quantum dots grown via theStranski-Krastanov techniques, are not well-described by thesingle-band particle-in-a-box approaches, despite the greatpopularity of such approaches in the experimentalliteratures.11,15,16,32 The model contains significant quantita-tive errors37 and also qualitative errors, whereby cylindri-cally symmetric dots are deemed to have, by symmetry, nofine-structure splitting, no polarization anisotropy, and nosplitting of �twofold degenerate� p levels and d levels, allbeing a manifestation of the “farsightedness effect.”30

III. RESULTS

Using the pseudopotential approach for single-particleand configuration interaction approach for the many-particlestep, we studied the electron or hole addition energy spec-trum up to 6 carriers in lens-shaped InAs dots embedded in aGaAs matrix. We study dots of three different base size, b=20, 25, and 27.5 nm, and for each base size, two heights,h=2.5 and 3.5 nm. To study the alloy effects, we also calcu-lated the addition spectrum for alloy dots In1−xGaxAs/GaAsof h /b=3.5/25 nm dots, with Ga composition x=0, 0.15,0.3, and 0.5. In this section, we give detailed results of thesingle particle energy levels and Coulomb integrals, and theaddition energy as well as ground state configurations. Wealso compare the results with what can be expected from theparabolic 2D-EMA model.

A. Single-particle level spacing: Atomistic versus 2D-EMAdescription

1. Electron levels

We depict in Fig. 2 the calculated energy-level diagram ofa pure lens-shaped InAs/GaAs quantum dot, with height h=2.5 nm and base b=20 nm. Figure 2 shows that the elec-tron confinement energy is 230 meV, somewhat larger thanthe hole confinement energy �190 meV�. The p levels aresplit as are the d levels, even though the dot has macroscopiccylindrical symmetry �see below�.

The pseudopotential calculated electron single-particleenergy spacings are summarized in Table I for QDs of dif-ferent heights, bases, and alloy compositions. Table I givesthe fundamental exciton energy EX calculated from CI ap-proach for each dot. These exciton energies are between 980and 1080 meV for pure InAs/GaAs dots, and can be as largeas 1297 meV for In1−xGaxAs/GaAs alloy dots. This rangeagrees very well with experimental results for these classesof dots, ranging from 990 to 1300 meV.8,11,38,39

�a� s-p and p-d energy spacing: From Table I, we see thatfor electrons in the lens-shaped dot, the s-p energy levelspacing �sp=�p−�s and p-d energy level spacing �pd=�d−�p are nearly equal, as assumed by the 2D harmonic model.The energy spacing �sp and �pd range from 50 to 80 meV�Fig. 2�, depending on the dot geometries. The electron en-ergy spacings decrease with increasing QD base sizes. Theelectron energy spacing of alloy dots are much smaller thanthose in pure InAs/GaAs QDs, because of reduced confine-

ment. For Ga rich dots �x=0.3–0.5�, the single-particle en-ergy level spacings range from 30 to 45 meV. These valuesagree with the infrared absorption measurements4,5 of intra-band transitions of alloy InGaAs QDs, which give �sp�41–45 meV. When the Ga composition reaches x=0.5, thes-p energy level spacing �sp becomes significantly differentfrom the p-d energy level spacing �pd, thus deviating fromharmonic potential approximation.

�b� Shell definition: Figure 2 shows that the energy levelsof electrons in a lens-shaped dot have well defined s, pd shellstructure. However, while effective mass and k · p modelspredict degenerate p and d levels, for cylindrically symmet-ric �e.g., lens-shaped� QDs, atomistic calculations show thateven in perfect lens-shaped dots, the p-p and d-d levels aresplit by 2–4 meV �Fig. 2� due to the actual C2v symmetry.We denote the two p levels as p1 and p2, and similarly, thethree d levels as d1, d2, and d3, in increasing order of energy.The results listed in Table I show that �pp=�p2

−�p1and �dd

=�d2−�d1

are very sensitive to the aspect ratio of the dotswhile not being very sensitive to the alloy compositions. Fig-ure 3 depicts �pp=�p2

−�p1versus dot heights �Fig. 3�a�� and

bases sizes �Fig. 3�b��. In general, we see that �pp increaseswith increasing dot height, and it decreases with increasingdot base size.

2. Hole levels

In contrast to electrons, hole single-particle levels �TableI� display a much more complicated behavior that is totallybeyond the EMA description.

�a� s-p spacing: As one can see from Table I, the hole

FIG. 2. The schematic energy-level diagram �in meV� of a purelens-shaped InAs/GaAs quantum dots, with height h=2.5 nm andbase b=20 nm. WLe and WLh denote the wetting layer energylevels for electrons and holes, respectively. The CBM and VBMcorrespond to the conduction band minima and valence bandmaxima of �unstrained� bulk GaAs. Ex is the excitonic transitionenergy.

MULTIPLE CHARGING OF InAs/GaAs QUANTUM DOTS¼ PHYSICAL REVIEW B 73, 115324 �2006�

115324-5

Figure 2.6: Energy levels for QDs calculated by different methods. Quantum-dot Linear absorptionspectra of InAs pyramid QDs of different sizes b (the arrows with “quantum numbers” indicate thestrongest contributing transitions) [31] (a), electronic spectrum for a InAs-GaAs dot with a base 124 A andheight 62 A [29] (b) and the schematic energy-level diagram (in meV) of a pure lens-shaped InAs/GaAsquantum dots, with height h=2.5 nm and base b=20 nm [28] (c).

QD1 QD2 QDM-1 QDM

...

ρ(E)

EQWe

QWh

ESe

GSe

GShESh

1063.1 meV

930.6 meV

873.0 meV

-73.0 meV

-85.4 meV

-186.9 meV

Figure 2.7: Energy levels with carrier transitions for a single quantum dot (in the center of theinhomogeneous distribution, as they are implemented in the model) and inhomogeneous distributionof M QDs with a common quantum well. ρ(E) represents the total density of states for each level.

In addition, carriers can not stay on a single energy level for an infinite time due to transitions,carrier-carrier scattering and interaction with phonons. This limitation of the dephasing time producehomogeneous broadening of the spectral line for a single QD energy level as presented in Fig. 2.4.

Due to the stochastic growth processes, all QDs have different shape and size, and different transitionenergies, as presented in Fig. 2.8. This inhomogeneous distribution of QDs is usually described by aGaussian (normal) distribution. Each QD subgroup (labeled as m) consists of two separate levels -ground state (GS) and excited state (ES). The fraction of m-th QD subgroup is labeled as D(m) andassumed to follow a Gaussian distribution:

D(m) =1

σGS,02√

2exp

((m−M/2)2

2σ2GS,0

)(2.5)

σGS,0 =ηGS,0

∆EGS,02√

2 ln 2

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f

f

ρ

ρ(a)

(b)

Figure 2.8: Schematic distribution of QDs in the inhomogeneous broadening for same QDs (a) anddifferent QDs (b). Left part present shapes and sizes of QDs, right part present corresponding density ofstates for this QD.

2 4 6 8 1 0 1 2 1 4 1 61 1 7 51 2 0 01 2 2 51 2 5 01 2 7 51 3 0 01 3 2 51 3 5 01 3 7 5

N u m b e r o f Q D - g r o u p

Reso

nanc

e Wav

eleng

th [nm

] G S E S

( a )

1 2 6 0 1 2 8 0 1 3 0 0 1 3 2 0 1 3 4 0 1 3 6 00 . 0 0

0 . 0 2

0 . 0 4

0 . 0 6

0 . 0 8

0 . 1 0

0 . 1 2( b )

Relat

ive De

nsity

of Sta

tes

R e s o n a n c e W a v e l e n g t h [ n m ]

Figure 2.9: Wavelengths distribution for GS and ES of QD groups (a) and the relative density of statesfor GS (b).

where σGS,0 is the normalized broadening relative to the energy separation between QD-groups, ηGS,0 isthe full width at half maximum (FWHM) of the inhomogeneously broadened distribution of the QD GS,∆EGS is the energy separation between different QD groups, and M is the total number of QD groups.

Figure 2.7 shows the energy levels for a single QD embedded in a QW and the associated carriertransitions (left) together with the scheme of the inhomogeneously broadened QD ensemble (right).Energy levels are presented for an injected current of 250 mA and essentially based on [23]. Figure 2.9presents the inhomogeneous distribution of 16 QDs as it is implemented in the model.

2.3 Quantum dot semiconductor optical amplifier

The experimentally available QD-SOA discussed here was produced by the workgroup of Prof. Dr. D. Bimberg.It is a 2 mm dot-in-a-well (DWELL) QD-SOA with 10 stacked InAs QD layers and a ridge waveguidestructure of width 4 µm and height 0.45 µm. It has a 2D QD density of 2·1015 m−2 per layer and anemitting wavelength of 1.3 µm. The barriers between the QD layers are p-doped with a hole density of4·1015 m−2 per layer in the active region. The facets are tilted by an 8◦ angle to reduce reflectivity. It isschematically presented in Fig. 2.10.

Experiments were provided in the Fraunhofer Heinrich Hertz Institute in the work group of Dr. ColjaSchubert [37] as a part of the research centre SFB-787 of the Deutsche Forschungsgemeinschaft (GermanResearch Foundation).

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Wafer

Insulator

Cladding

MetalContact

Active Region

10 × QD LayersPin

Poutpn

(a) (b)

Figure 2.10: Schematic illustration of the QD-SOA structure (a) and setup with injection current andoptical power (b).

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2.4 Carrier dynamics

Carrier dynamics are one of the most important part of the QD-SOA modeling and of this work. Itdetermines the material gain dynamic and as a result the optical amplification.

Carrier dynamics involve different kind of processes, like Auger-like carrier-carrier scattering [3,20,38–41,41–47] or carrier-phonon interactions [3,39,47–49]. Carrier-phonon interaction is the main relaxationmechanism in the bulk structures, but it is significantly limited in QDs due to the discrete energystructure, as the energy separation between levels is not equal to the energy of optical or acoustic phonons.This so called “photon bottleneck” increases the carrier transition times for QDs as compared to the bulkstructures. So carrier-phonon interaction is not included in this model, as is does not play an importantrole, but it may be required in the future for more detailed observation.

In this work we are considering only Auger-like carrier-carrier scattering, which produce carriertransitions between levels [20]. All included types of carrier-carrier scattering are schematically presentedin Fig. 2.11 for electrons in the conduction band. Included processes for holes in the valence band areanalogeous.

Auger-like carrier scattering processes include carriers in the quantum well, so the transitions ratesare functions of the QW carrier density. These functions in the model are used as a dependence ofelectron QW-density for electrons or holes QW-density for holes for simplicity. In this case we supposethe constant relation between electrons- and holes- QW populations. For a more detailed observationit is required to calculate the transition rates based on QW-carrier densities in conduction and valencebands separately.

The calculated transition rates do not include energy difference between the QD-groups in theinhomogeneous distributions, so all QDs for a single QD-SOA segment have the same transition rates.This approximation can be used for injected current over 125 mA/mm (for the experimentally availableQD-SOA) as we have relatively high carrier density in the QW and high transition rates. But separatetransition rates for different QD-groups must be implemented in the model for a small injected currentand/or low QW-densities as this effect produces the blue shift, responsible for a longer wavelength of themaximum GS amplification in the QD-SOA.

The blue shift leads to different carrier distributions over the inhomogeneous broadening and thusproduces higher gain and SE in the blue part of the spectra, as it will be presented in the amplifiedspontaneous emission spectrum in section 5.1. For a negligible QW carrier density we have significantlyhigh importance of the capturing from QDs with high transition energies, compared to the QDs withlow transition energies, because the energy separation between QDs and QW levels is smaller for QDswith larger transition energy. Therefore carriers in these QDs have more chances to escape back to thereservoir, while escaping from dots with small energy is negligible and carriers can stay there for a longertime.

The approximations used to calculate the transition rates are presented in the App. B.

Calculated rates are presented in Fig. 2.12. As visible, the transition rates for carrier capturing are

QD

QW

QW

SQW-GSe SQW-ESe SES-GSe

Figure 2.11: Included types of carriers transitions due to carrier-carrier scattering for electrons in theconductive band for carrier transition rates between QW and GS SeQW−GS , between QW and ES SeQW−ESand between ES and GS SeES−GS .

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increasing in almost square dependence, as for high carrier density we have more carriers in QW, thatcan be captured, and more partners, that can be excited in the QW to release energy.

0 1 2 3 4 5 6 7 8 9 1 00 . 0

0 . 2

0 . 4

0 . 6

0 . 8 ( a )( a )

S [ps

-1 ]

n Q W [ 1 0 1 5 m - 2 ]

S eQ W - E S

S eQ W - G S

S hQ W - E S

S hQ W - G S

0 1 2 3 4 5 6 7 8 9 1 00 . 0

0 . 5

1 . 0

1 . 5

2 . 0

2 . 5( b )

S [ps

-1 ]

n Q W [ 1 0 1 5 m - 2 ]

S eE S - G S

S hE S - G S

Figure 2.12: Capturing from QW into QD levels (a) and intradot relaxation (b) rates as they areimplemented in the model for different QW carrier densities [20].

For intradot transitions (between ES and GS) relaxation rates have linear dependence on the QWdensity for small QW carrier densities, as more carriers in QW can participate in this process. While,for further increasing of the QW carrier density, these rates have smaller increase or can even decrease(as visible for holes). This is due to dramatically reducing of the free space on lower QW energy levels,which are occupied by carriers.

All carriers in the QW are assumed to have the Fermi-Dirac distribution all the time, so carrier

escaping rates from QD levels into QW (Se(h)ES−QW and S

e(h)GS−QW ) can be calculated based on capturing

rates (Se(h)QW−ES and S

e(h)QW−GS) as [20]:

Se(h)ES−QW = S

e(h)QW−ES exp

(−

∆Ee(h)QW−ES

kBT

)(exp

(ne(h)QW

kBTme(h)eff /(π~)2

)− 1

)−1

Se(h)GS−QW = S

e(h)QW−GS exp

(−

∆Ee(h)QW−GS

kBT

)(exp

(ne(h)QW

kBTme(h)eff /(π~)2

)− 1

)−1

where ∆Ee(h)QW−ES and ∆E

e(h)QW−GS are energy separations between QW and intradot levels (ES and GS

respectively) for electrons and holes, me(h)eff are effective masses for electrons and holes and T is the carrier

temperature.

The intradot exciting rates Se(h)GS−ES can be calculated similarly:

Se(h)GS−ES = S

e(h)ES−GS exp

(−

∆Ee(h)ES−GSkBT

)(2.6)

where ∆Ee(h)ES−GS is the energy separations between ES and GS levels for electrons and holes.

The carrier dynamics are described by a set of coupled rate equations [22] for the 2D QW densities

ne(h)QW and the occupation probabilities of the intradot GS and ES levels, f

e(h)GS (m) and fe,hES (n), respectively.

Time t and length z indices are omitted in these equations for a simplicity. P-doping is included via anadditional number of holes in each segment [22,50].

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∂ne(h)QW

∂t= ηInj

Iinje

dz

L−BQWneQWnhQW (2.7)

−∑m

Se(h)QW−ES(m)(1− fe(h)

ES (m))εESD(m)D2D

+∑m

Se(h)ES−QW (m)f

e(h)ES (m)εESD(m)D2D

−∑m

Se(h)QW−GS(m)(1− fe(h)

GS (m))εGSD(m)D2D

+∑m

Se(h)GS−QW (m)f

e(h)GS (m)εGSD(m)D2D

∂fe(h)ES (m)

∂t= S

e(h)QW−ES(m)(1− fe(h)

ES (m)) (2.8)

− Se(h)ES−QW (m)fe,hES (m)

− Se(h)ES−GS(m)f

e(h)ES (m)(1− fe(h)

GS (m))

+ Se(h)GS−ES(m)f

e(h)GS (m)(1− fe(h)

ES (m))εGSεES

− feES(m)fhES(m)

τES− RstimES (m)

D2DεGS(ES)D(m)WdzNlay

∂fe(h)GS (m)

∂t= S

e(h)QW−GS(m)(1− fe(h)

GS (m)) (2.9)

− Se(h)GS−QW (n)fe,hGS (n)

+ Se(h)ES−GS(m)f

e(h)ES (m)(1− fe(h)

GS (m))εESεGS

− Se(h)GS−ES(m)f

e(h)GS (m)(1− fe(h)

ES (m))

− feGS(m)fhGS(m)

τGS− RstimGS (m)

D2DεGS(ES)D(m)WdzNlay

where Iinj is the injected current, ηInj is the injected current efficiency, L is the device length, BQW ,τGS and τES are spontaneous recombination coefficients and εGS(ES) are the density of states for GS(ES)

level. Se(h)QW−ES , S

e(h)ES−QW , S

e(h)QW−GS and S

e(h)GS−QW are carrier capture (escape) rates from (to) the QW.

Se(h)ES−GS and S

e(h)GS−ES are intradot carrier transition rates. Scattering rates are calculated here based

upon Coulomb scattering within a microscopic theory [20].The calculation of the stimulated emission RstimGS(ES)(m) for the m-th QD group in Eq. 2.8 and Eq. 2.9

depends on the difference between output | EF (R),GS(ES),out(m) |2 and input | EF (R),GS(ES),in(m) |2 photondensities for the corresponding m-th filter, thus it is possible to separate amplification from each QDsgroup from the total signal amplification. The stimulated emission rate obtained is then given by:

RstimGS(ES)(m) =(| EF,GS(ES),out(m) |2 − | EF,GS(ES),in(m) |2 +

+ | ER,GS(ES),out(m) |2|2 − | ER,GS(ES),in(m) |2)

(2.10)

To receive better agreements with experimental results it was required to compute injection currentefficiency as a function of the injection current:

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ηInj = 0.35 + 0.25IInj − 250mA

250mA(2.11)

These approximations are acceptable for injected currents IInj in the range 250-500 mA. Increasingthe injected current efficiency for higher current sounds strange, but it can be explained as a saturationof e. g. the nonradiative processes, which decrease with electric pumping.

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Chapter 3

Time-domain modeling

3.1 Time-domain modeling fundamentals

This model is operating with a slowly varying complex envelope of the electric field E(t) around the centralfrequency f0 of the model. Corresponding to the Nyquist−Shannon sampling theorem for a calculationtime step dt, it is possible to model any signal in the frequency range (f0 − 1/2dt, f0 + 1/2dt]. Theselected time step dt=20 fs allows to process the optical signal in 1/dt=50 THz spectral range. This isenough to model the amplified spontaneous emission from GS and ES levels, as well as ultrashort ∼100 fspulses. A short calculation step is also necessary to have the correct filter behavior, as their FWHM (2.4THz) should be much smaller than the total calculated spectral range.

In the most trivial case the slowly varying complex envelope of the signal E ′(t) can be described as afunction of the total optical power P (t) and the phase ϕ(t):

E ′(t) =√P (t) · eiϕ(t)

P (t) =|E ′(t)|2

Instead of the normalized optical power (electric field), processing by the photon density flow makessimulations much more simple, as to calculate stimulated recombination it is required to know the numberof photons, not their optical power. For broadband signals we will have significant energy difference forphotons with different frequencies, which will require to recalculate “optical power→ number of photons”on each calculation step. In the presented model the transformation from the photon density to the opticalpower is provided only once, after the simulation, to analyze the optical power.

In this case the chirp-free narrowband amplitude modulated signal with a central frequency fS andtime-dependent optical power P (t) can be described in the model with central frequency f0 as:

E(t) =

√P (t)

hfS· ei2π(fS−f0)t

P (t) =|E(t)|2hfS

For two signals with powers P1(t) and P2(t) with central frequencies f1 and f2, the total signal is:

E(t) =

√P1(t)

hf1· ei2π(f1−f0)t +

√P2(t)

hf2(· ei2π(f2−f0)t (3.1)

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f

g( f )

f

(a)

g( f )

(b)

Figure 3.1: A comparison of the frequency dependences of the gain for conventional (a) and presented(b) models. Amplification from a single QD is a flat black line.

3.2 Conventional time-domain models

The usual way to model a QD-SOA is to separate all QDs into multiple groups in dependence to theirresonance transition energies, corresponding to the inhomogeneous broadening. The energy distancebetween different QD groups is smaller than the homogeneous broadening to receive a good gain overlapfrom different dots. This method allows us to calculate frequency dependence of the gain. Carrierdynamics (i.e. gain dynamics) for each QD-group are calculated separately [5–24]. Schematically thedistribution of QD energy levels and their interaction with QW carrier reservouir is presented in Fig. 2.7.

To model a light propagation through the QD-SOA, the total input signal is divided into manyseparate time-domain (TD) models with different central frequencies, i.e. “spectral resolution” for thesignal is created. Each spectral component is calculated separately in this case. The material gain for eachtime-domain model is calculated as a summation over material gain provided by each QD group. Thesetime-domain models can interact with each other only through the material gain (cross-gain modulation,XGM) and phase-based processes are not possible between them [5–24].

So if we have two or more optical signals with different central frequencies, each of them create carrierdensity modulations for all carrier groups, which leads to the gain modulation for all others spectralcomponents of the input signal and makes it possible to model a correct cross-gain modulation in theQD-SOA.

But this realization has two main disadvantages:- it does not include frequency dependence of induced phase modulations, created by carrier densities

modulations, and does not include cross-phase modulations between optical signals.- all frequency component inside one time-domain model have the same gain value. This is not

important for CW signals or broad pulses that have very narrow spectrum, but leads to a dramaticinaccuracy for subpicosecond pulses, as their spectra can be as broad as 10 THz (55 nm for a centralwavelength of 1310 nm). This width of the spectrum is comparable to the inhomogeneous broadening ofthe gain spectrum.

To provide more accurate modeling it is required to simulate very broadband signals and all spectralcomponents should be treated in one single time-domain model to include phase-based effects correctly,and the material gain should be frequency dependent. The same kind of models were implemented for abulk SOA [51,52], but was not yet created to operate with the QD-SOA.

To reach this aim we use digital filters [53] to simulate the frequency dependence of the gain for eachQD group. Each filter has a Lorentzian line shape with a finite width corresponding to the homogeneousbroadening of the QDs. This allows to create a continuous correct gain profile over a very large rangeof frequencies compared to a conventional model where gain is calculated only for a central frequency ofeach time-domain model. Gain profiles are schematically presented in Fig. 3.1.

Figures 3.2 and 3.3 show the main differences between the conventional model and the presented one.This pictures show a material gain and signal spectra for a propagation of ultrashort pulses and a four-wave mixing process. In the conventional model there is the same amplification for all spectral componentinside a single time-domain model. This makes it not appropriate for signals, which are broader thanthe homogeneous broadening (2.4 THz) and completely pointless for signals, which are broader than theinhomogeneous broadening (8.5 THz). Another disadvantage is that the all spectral components inside

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f

(b)

g( f )

f

g( f )

(a)

f

P( f )

f

P( f )

P( f )

f

Figure 3.2: A comparison of the conventional model with the presented one for an amplification ofultrashort pulses with broad spectrum. Calculated gain profiles are presented above and signal spectraare below. For a conventional model the whole signal is located in one time-domain model and thestimulated recombination is calculated if only all photons have the same frequency, equal to the centralfrequency of this time-domain model. In this case we will receive a significant spectral hole burning andthe same gain for all spectral components. The presented model (b) aviods these defects. It can provideboth correct gain saturation (stroke line represent unsaturated gain) and correct amplification for allspectral components.

f

g( f )

f

(a) (c)

g( f )

f

g( f )

(b)

f

P( f )

f

P( f )

f

P( f )

P( f )

f

Figure 3.3: A comparison of the conventional model with the presented one for a FWM simulation.Calculated gain profiles are presented above and the signal spectra are below. In a conventional modelboth input continuous waves can be located in two different time-domain models (a) or in a single one(b). In the first case, we well obtain correct frequency dependence of the gain, but absence of the FWMproducts. In the second case we will have FWM products, but also a wrong gain profile. The presentedmodel (c) allows to have both correct FWM products and the correct gain profile.

a single time-domain model create a stimulation recombination only for a central frequency of this TDmodel, which produces wrong spectral hole burning, as schematically presented in Fig. 3.2a and 3.3(a,b).If two signals will be located in different time-domain models, as presented on Fig. 3.3a, we will loseall phase interaction between them, as the only connection between signals is the material gain, whichdepends on the optical power only. These disadvantages are fundamental and cannot be eliminated byusing local improvements.

As we can create a frequency dependence of the gain by utilizing the digital filters, we can avoid allthese defects and significantly simplify the model. We can use only a single broadband time-domain nodelto hold all spectral components. The material gain from each QD group is included as a filter whichamplify spectral components that should be amplified based on the homogeneous broadening of this QDs.As a bonus, we additionally obtain a correct phase modulation from the gain modulation for all frequencycomponents. As visible in Fig. 3.2b and 3.3c the presented model is free from the disadvantages describedabove and surpasses a conventional model.

In a conventional model, if both CW signals are located in different time-domain models (as presentedin Fig. 3.3a) we observe almost a correct gain profile, but no FWM products in the output spectrum, asin this case we have neither gain pulsation nor XPM. Otherwise, if both input CW signal are located ina single time-domain model (as presented in Fig. 3.3b), we have FWM products on the output. But as

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the gain is calculated correctly only for a central frequency of this time-domain model, we observe wrongtotal gain profile and both signals and FWM products have the same amplification.

Figure 3.3c schematically present a FWM simulation with the presented model. As both input signalsare inside one TDM, we have correct XPM and FWM products. Due to digital filters we can achieve acorrect frequency dependence of the gain for all spectral components.

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3.3 Propagation Equation

The propagation equation describes the light propagation through a semiconductor medium. It ispresented in the differential form for the electric field and calculated for each z segment of the QD-SOA separately.

The material gain is included in the model by using digital filters, and the influence of all other effectsis described in the propagation equation for the optical signal. For clarity only the forward traveling wave(subscript F ) is described. The backward traveling wave (subscript R) can be described in an equivalentway.

∂EF∂z

=

[−1

2αint −

(γTPA2

+ ib2

)(|EF |2 + |ER|2)~ωS − i

ω

c0∆nFCA

]× EF + (3.2)

+

M∑m=1

hm ∗ EF + ESE

where αint is the waveguide internal loss, γTPA is the two-photon absorption coefficient, b2 is the phasemodulation parameter associated with TPA and the Kerr effect, ωS is the central angular frequencyof the signal, ∆nFCA is the refractive index change due to free-carrier absorption, and ESE(t, z) is thetotal spontaneous emission from all QDs. Asterisk ∗ designates a convolution between signal EF and theimpulse responses from the filters hm describing the gain of the m-th QD-group in one segment.

All these effects are described below in more details.

3.4 Internal losses, two-photon absorption,Kerr effect and free-carrier absorption

The internal waveguide losses yield a decrease of the optical power due to a scattering, inner reflections,waveguide imperfection, absorptions (that does not involve carriers in the active area) and other minoreffects. They are usually defined by a linear coefficient αint. For a QD-SOA this coefficient usually hasa value of 150 m−1 for undoped QD-SOA, for a significant p-doping it is increased to 450 m−1 due toan increased free-carrier absorption in the active region [3, 50, 54]. These dopping induced carriers aresupposed to have a constant density and are not included in the “active carriers”, so they play no rolefor the calculated carriers dynamics.

The influence of the internal losses on the optical signal is included in the model by following simpleequation:

∂EF∂z

= −1

2αintEF (3.3)

The two-photon absorption (TPA) is an additional nonlinear loss mechanism in the semiconductormedium. It exist in the QWs, bulk barriers between layers and around the active region [55, 56]. Thiseffect has no interaction with QDs, so it is calculated in the same way, as for bulk- and QW-SOAs. Itcan be described by a coefficient γTPA:

∂EF∂z

= −(

1

2γTPA

)(|EF |2 + |ER|2)~ωSEF (3.4)

Total value of absorption is nonlinear and depends on the optical power. So the TPA is most importantfor short pulses when the peak power can reach 1 W and more. This effect can be neglected for CWsignals or NRZ-pulses, but it becomes very important for RZ-signals.

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A nonlinear phase modulation term b2 includes two physical process - the Kerr effect and the phasemodulation due to TPA. For most of the simulations both effects can be merged and included in onecoefficient. The final equation for the TPA and the Kerr effect is:

∂EF∂z

= −(

1

2γTPA + ib2

)(|EF |2 + |ER|2)~ωSEF (3.5)

The free carrier absorption (FCA) occurs due to absorption of free carriers and can be calculated witha Drude coefficient [18,57,58]:

CDrude = − e2

2ηbgε0m∗ω2S

(3.6)

where ηbg is the background refractive index, m∗e,h is the effective carrier mass. The value of inducedabsorption is negligible and is not included in the model however this effect leads to an induced changein the refractive index and to a phase change given by:

∆ηFCA =∑

N3De,hCDrude (3.7)

∆ϕFCA∂z

= −∆ηFCAωηbgc0

where Ne,h3D is the 3D QW carrier density in the active region.

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3.5 Diagram of the presented time-domain modeling

In the presented model the whole QD-SOA is divided into multiple segments in the length direction. Thelength of each segment depends on the calculation time step dt as

dz =c0ηbg

dt (3.8)

where ηbg is the background group index.For each z-th segment and t-th time moment, we have initial carriers distributions and optical signals,

which enter this segment from both directions (for forward EF and reverse ER propagations).The main aspects of the presented model are shown in Fig. 3.4. It consists of two main parts - the

carrier dynamics and the signal propagation.

Based on the initial carrier distributions in the quantum well ne(h)QW (t, z) and the occupation probabilities

fe(h)GS(ES)(m, t, z) of the QD levels, which are separate for electron (e) end hole (h) dynamics we can

calculate the carrier injection in a segment due to the injected current, spontaneous recombinations andintraband carriers transitions.

The occupation probability for QD levels are used to calculate values of the material gain g(m, t, z)from each QD group and calculate the signal amplification. Stimulated recombination is calculatedbased on the signal amplification. Spontaneous recombinations in QDs provide information about thespontaneous emission, which is added to the propagated signal.

After that, additional effects (internal losses, FCA and TPA) are applied to the signals.Final carrier distributions for each time step are used as the initial distributions for the same segment

for the next time step. Final optical forward (reverse) propagated signals are used as the input signalsfor the next (previous) segment of the QD-SOA for the next time step.

23

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Signal Propagation

Injected currentTransitions

Stimulated recombination

Spontaneous recombination

Filtering (opical gain)

Spontaneous emission

Internal lossesTPA, FCA

z

dt, dz=dt·c0/ηbg,

0 L

fES,GS (m,t,z) e,hnQW (t,z) e,h

EF (t,z)ER (t,z)

gES,GS (m,t,z)

RES,GS (m,t,z) stim

RES,GS (m,t,z) spont

fES,GS (m,t+dt,z) e,h

nQW (t+dt,z) e,h

EF (t+dt,z+dz)ER (t+dt,z-dz)

EF (t,L)ER (t,L)

EF (t,0)ER (t,0)

Carrier dynamics

Figure 3.4: The schematic diagram of the the bidirectional light propagation EF (R) through the SOAsegments dz (above) and the calculated effects in each segment for one time step dt (below). Calculated

carrier dynamics is based on the initial carrier distribution (ne(h)QW and f

e(h)GS(ES)) for this segment. Carrier

dynamics and optical signal segments interact with each other through the material gain gGS(ES),

stimulated recombination RstimGS(ES) and spontaneous recombination RspontGS(ES). Final carrier distributions

(ne(h)QW and f

e(h)GS(ES)) are used as initial for the same segment and next time step t+ dt, final values of

signals EF (R) are used as initial for the next z + dz (previous z − dz) segment for the next time step.

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Chapter 4

Digital Filters

4.1 Introduction

Digital filters perform simple arithmetical (linear) operations with discrete signals [53]. They are widelyused in telecommunication signal processing because they allow to change a spectrum of time domainsignal without its Fourier transformations into frequency domain, i.e. they are working as frequencyfilters [51,52].

Each filter is described by its transfer function H(f), where f is the discrete frequency. FunctionH(f) determinates the amplification for any frequency f of the input signal. For any input signal with adiscrete spectrum X(f), the spectrum on the output of the filter Y (f) is determined as a multiplication:

Y (f) = H(f)×X(f) (4.1)

As we have input x(t) and output y(t) signals in the time domain, we should use the impulse responsefunction of the filter h(t) (that is the Fourier transformation of the frequency transfer function) anda convolution operator ∗ instead of multiplication. Convolution of two functions in the time domainis equivalent to the multiplication in the frequency domain, but it does not require additional Fouriertransformations. In the time domain equation 4.1 has a form:

y(t) = h(t) ∗ x(t) (4.2)

where:

h(t) ∗ x(t) =

∫ −∞−∞

h(t− τ)x(t)dτ (4.3)

To explain the filtering process let us describe a filtering of a simple signal by a single filter. Theinput signal is two pulse sequences with different central frequencies:

Ein(t) = E1(t)exp(i2π(f1 − f0)t) + E2(t)exp(i2π(f2 − f0)t) (4.4)

where E1(2)(t) are the envelopes of both signals, f1(2) are the central frequencies of these signals and f0

is the central frequency of the time-domain model.Both signals are sequences of sech-shape pulses:

E1(t) = E2(t) = sech

(−|t− t0|

T

)(4.5)

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0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 00 . 0 0

0 . 2 5

0 . 5 0

0 . 7 5

1 . 0 0 ( a )

Ampli

tude

T i m e [ a . u . ]0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0

0 . 0 00 . 2 50 . 5 00 . 7 51 . 0 01 . 2 51 . 5 01 . 7 52 . 0 0 ( b )

Ampli

tude

T i m e [ a . u . ]- 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0

0 . 0 0

0 . 0 5

0 . 1 0

0 . 1 5

0 . 2 0

0 . 2 5 ��

Magn

itude

����� ��������������� p ����������

Figure 4.1: Power P1(2) of input pulses (a), power of the total signal Pin(t) on the input of filter (b) andits Fourier transformation (spectrum) (c). As visible, beatings in the total input signal were created bya summation over two input signal with different central frequencies.

- 2 0 0 2 0 4 0 6 0 8 0 1 0 00 . 0 0

0 . 0 1

0 . 0 2

0 . 0 3

0 . 0 4

0 . 0 5

0 . 0 6( a )

Ampli

tude

T i m e [ a . u . ]- 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0

0 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0 ��Ma

gnitu

de

������� ���� �� ������ p �������� �0 1 0 2 0 3 0 4 0 5 0

0 . 0 0

0 . 0 2

0 . 0 4

0 . 0 6

0 . 0 8

0 . 1 0

0 . 1 2

Ampli

tude

T i m e [ a . u . ]

Figure 4.2: Impulse response of the filter (a), the magnitude response (Fourier transformation of theimpulse responce) (b) and the convolution between input signal and pulse response of the filter (c). Asvisible, pulse responses are shifted to each time moment and multiplied by the values of the input signalfor this moments. Black line marks out a filter reaction on the signal at a single time moment.

where t0 is the central position of the pulses and T determine the broadening. The input signal ispresented in Fig. 4.1

The described filter has a Lorentzian shape in the frequency domain and centered by the centralfrequency of the time-domain model. In the time domain the filter is described by its pulse response.The pulse response of the filter is the Fourier transformation of its magnitude response. It describes thereaction of the filter (output signal) on the instantaneous ultrashort impulse (i.e. Dirac function) on theinput. The pulse response of discussed filter with zero central frequency has an exponential dependence:

h(t) = exp

(− t

T2

)exp

(dt

T2

)(4.6)

where T2 is parameter to determine the broadening of the filter. The exp(dt/T2) term were added tonormalize the amplification.

Impulse response of the filter and its frequency dependence (magnitude resonce) are presented inFig. 4.2.

To find the output signal after the filtering it is required to find the convolution between input signaland pulse response of the filter. For a discrete signal Pin(t) its convolution with the filter h(t) is describedas:

Ein(t) ∗ h(t) =

∞∑τ=0

E(t)h(τ − t) (4.7)

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0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 00 . 0

0 . 1

0 . 2

0 . 3

0 . 4

0 . 5 ( a )

Ampli

tude

T i m e [ a . u . ]- 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0

0 . 0 0

0 . 0 5

0 . 1 0

0 . 1 5

0 . 2 0

0 . 2 5 ��

Magn

itude

������� ���� �� ������ p �������� �

Figure 4.3: Magnitude Pout(t) of the output signal after the filtering (a) and its spectrum (b).

So, for each sample of the input signal, the impulse response is shifted to this time moment andmultiplied by the value of the signal at this time moment. The total output signal is a summation overall pulse responses for all time moments. Figure 4.2c describes this process.

The output signal after the filtering is presented in Fig. 4.3. As visible, signal frequencies around thecentral frequency of the filter are almost the same, while all other spectral components are significantlysuppressed. So we have changed the spectrum of the time-domain signal without its transformation intofrequency domain.

This process is schematically presented in Fig. 4.4. Filtering of the signal in the time-domain makesthe same output as the multiplication of the signals spectrum with the magnitude response of the filter.

4.2 Filter equation in Z space

Digital filters can be expressed more easily with a transfer function in the Z-domain [53] (by the Z-transformation of the H(f) function) with complex coefficients Am and Bn:

H(Z) =

B0 +N∑n=1

BnZ−n

1 +M∑m=1

AmZ−m(4.8)

This coefficients determinate the response of the filter on the complex exponents with frequencies:

Z−n = exp (−iωndt)

where dt is the time step of the modeled signal.Structures of the infinite impulse responce filters of the first order are shown in Fig. 4.5. A is a filter

coefficient that specifies the spectral width of the filter and its central frequency, B is the filter coefficientthat specifies the amplification, G is the material gain in the current time moment. The input signal forthe current time step enters the delay element and the summator, where input and output signals fromdelay elements (for the previous time step) are added to it. The signal from the summator is the outputsignal of the filter.

It can be easily seen, what for any discrete input signal x(t), the output signal y(t) is determined as:

y(t) = B0x(t) +

N∑n=1

Bnx(t− ndt)−M∑m=1

Amy(t−mdt) (4.9)

where Bn and Am are constant coefficients.

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FFT FFT

×

=

=

FFT

Figure 4.4: Interconnection of the filtering time-domain signal and changing of its spectrum in thefrequency domain. Upper figures describe a convolution of the time-domain signal and pulse response ofthe filter, lower figures describe multiplication of the signal spectrum and the filter magnitude response.Each element of the transformation in the time-domain is interconnected to the corresponding elementof the frequency-domain transformation by a fast Fourier-transformation.

4.3 Structures of gain filters

Carrier-photon interaction (single photon absorption or emission) in a semiconductor medium have aLorentzian line-shape of the spectrum and it is shown in Fig. 2.4. To model it, we are using an infiniteimpulse response filters of the first order. Response function of this filter is:

H(Z) =B(G− 1)

1−AZ−1(4.10)

To operate with a complex envelope we use filter equations in time-space:

Eout(m, t) = B(G(m, t)− 1)Ein(m, t) + ZG(m, t− dt)ZG(m, t) = A(m)Eout(m, t) (4.11)

where Ein(m, t) is the input complex signal of the m-th filter for t-th time moment, Eout(m, t) is theoutput complex signal of the filter for t-th time step, G(m, t) is the peak filter magnitude (material gain)and ZG(m, t) is the delay element. For each QD group the coefficients A and B are calculated as:

A(m) = exp

(i2π(fF (m)− f0)dt− dt

T2

)(4.12)

B = 1− |AG| = 1− exp(− dtT2

)(4.13)

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B(G-1)

A

Σ

Z-1

Input InputOutput Output

(a) (b)

A

Z-1

Figure 4.5: Filter structures for the material gain (a) and the spontaneous emission (b). A and B are filtercoefficients and Z−1 is the delay element. Both filters have the same form but different coefficients. Thedifference is a multiplication by material gain factor (G−1) for the calculation of the signal amplification.

where f0 is the central frequency of the model, fF (m) is the resonant transition frequency of m-th QDgroup and T2 is the dephasing time.

To obtain the required amplification and phase shift for one length step, the signal “propagates”through all filters (all QDs) as shown in Fig. 4.6 and the amplified signal is created on the output ofthe summator. Each filter amplifies spectral components of the signal near central filter frequency, soamplification by multiple filters (with different central frequencies) produce the same result as amplificationof the signal by multiple QDs with the inhomogeneous distribution.

Magnitude and phase responses of these filters are shown in Fig. 4.7. These filters amplify opticalsignals around resonance frequency and create negligible signal change for the other frequencies. Figure 4.8presents the signal on the output of the summator (the sum of the filtered and the input signals) for asingle filter.

It is visible, that by applying the Lorentzian shape to the magnitude, we automatically get the requiredphase modulation as it should be corresponding to Kramers-Kronig relations [59–63] i.e. it is not requiredto add any more parameter (like alpha-factor) to obtain the required phase or refractive index modulation.In other words, if we change a filter gain (have a gain pulsation) this will automatically change the opticalphase, this is equal to the utilization of the α-factor. This gain-phase relation is the fundamental forany wave (electro-magnetic, acoustic etc.) and can be used as a verification of any model. If the phasemodulation is correct for any signal frequency, this means that the frequency dependence of the gain isalso correct. While if additional coefficients for phase modulation (like alpha-factor) are required, thismeans that the model based on simple approximations and may be used only for most simple behaviors,like CW lasers, which have constant CW optical power with very narrow spectrum.

Figure 4.10 presents a relation between an amplification by a single filter and multiple filters with theinhomogeneous distribution of their frequencies and peak gains.

Visible imperfection on frequencies far away from the resonance (like additional gain and positivephase modulation in the region +20...+25 THz) are created by a frequency limitation of the model (50THz) and can be improved by using a shorter time step. The principle of discrete signals postulates anydiscrete signal has has periodic spectrum. The same can by applied to the filter responses. Figure 4.3shows magnitude and phase responses of the filters with the same coefficients, but different calculationtime steps (20 fs. and 5 fs.). As visible, both of them have the almost identical responses aroundresonance frequency, but they a significantly different for detunings of more then ±20 THz, due to itsperiodic structure.

Making simulations for more broadband models does not create any visible change in the results. Sothis time step was selected as a compromise between useful precision and the reducing of the calculationtime.

Three main characteristics of this filter (and corresponding physical processes) are:- central frequency (resonance transition frequency), fF- peak magnitude response (maximum gain), G- filter broadening (homogeneous broadening, dephasing time) T2.

29

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f

E(f)

Filter 2Filter 1 Filter 3

E(t)

f

EF(f)

E2(t)

E(t+dt)

f

G3(f)

f

G2(f)

f

G1(f)

E1(t)

SE

E (t)

f

f

E(f)

E3(t)

int. los.

TPA

FCA

SE E (f)SE

+EF(t)

+ +

+

Figure 4.6: The principle of filter-based signal amplification for one length and time step. Left partshows the signal processing schematically, the right part represents corresponding power spectrum: E(t) -input signal for current length step and Em(t) - signal on the output of m-th filter, ESE(t) - spontaneousemission from QD groups, and E(t + dt) - output signal for the current step and input signal for nextlength step and time moment. Dashed line in the right part of the figure shows spectrum before applyingfilters or spontaneous emission, respectively, for comparison.

The central frequency is the resonance transition frequency of the electron-hole recombination. In themodel (in the slow-varying envelope approximation) we are using relative frequencies fF = ftransition−f0

around central frequency f0.Peak filter magnitude G correspond to the linear amplification of the each QD group:

GGS(ES)(m) = exp

(1

2gGS(ES)(m)dz

)gGS(ES)(m) = ΓagD(m)εGS(ES)Nlay

× (feGS(ES)(m) + fhGS(ES)(m)− 1)

Here, g [m−1] is the linear amplification, dz - length of one SOA segment, factor 1/2 comes from theelectric field calculation instead of optical power, Γ is the optical confinement factor, ag is the linear gainparameter, εGS(ES) is the degeneracy of the GS (ES) levels of the QDs, and Nlay is the number of QDlayers.

Full-width at half-maximum (FWHM) of the filter is specified by the time scale parameter T2 (dephasingtime):

T2 =h

πγ0(4.14)

where γ0 is the FWHM of homogeneous broadening energy. The value of homogeneous broadening wereselected to 10 meV for a room temperature as a typical value from multiple publications [43,64–77]

Spontaneous emission filters

To receive a Lorentzian spectrum of spontaneous emission, the white noise propagates through the SEfilters. SE filters have a form:

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- 2 0 - 1 0 0 1 0 2 00 . 0 0 0 00 . 0 0 0 10 . 0 0 0 20 . 0 0 0 30 . 0 0 0 40 . 0 0 0 50 . 0 0 0 6 ( a )

f = 2 1 9 . 4 T H z ( - 1 7 . 7 T H z ) , G = 1 . 0 0 0 4 f = 2 2 5 . 4 T H z ( - 1 1 . 7 T H z ) , G = 1 . 0 0 0 6

Magn

itude

R e l a t i v e F r e q u e n c y [ T H z ]

2 2 0 2 3 0 2 4 0 2 5 0 2 6 0A b s o l u t e F r e q u e n c y [ T H z ]

- 2 0 - 1 0 0 1 0 2 0- 1 . 5

- 1 . 0

- 0 . 5

0 . 0

0 . 5

1 . 0

1 . 5 f = 2 1 9 . 4 T H z ( - 1 7 . 7 T H z ) f = 2 2 5 . 4 T H z ( - 1 1 . 7 T H z )

A b s o l u t e F r e q u e n c y [ T H z ]

Phas

e [rad

]

R e l a t i v e F r e q u e n c y [ T H z ]

2 2 0 2 3 0 2 4 0 2 5 0 2 6 0( b )

Figure 4.7: Calculated magnitude (a) and phase (b) responses of digital filters with transition frequenciesof 219.4 THz and 225.4 THz (relative frequencies in the model are -17.7 THz and -11.7 THz) and gain0.0004 and 0.0006 per length step.

- 2 0 - 1 0 0 1 0 2 01 . 0 0 0 01 . 0 0 0 11 . 0 0 0 21 . 0 0 0 31 . 0 0 0 41 . 0 0 0 51 . 0 0 0 6

f = 2 1 9 . 4 T H z ( - 1 7 . 7 T H z ) , G = 1 . 0 0 0 4 f = 2 2 5 . 4 T H z ( - 1 1 . 7 T H z ) , G = 1 . 0 0 0 6

A b s o l u t e F r e q u e n c y [ T H z ]

Magn

itude

R e l a t i v e F r e q u e n c y [ T H z ]

2 2 0 2 3 0 2 4 0 2 5 0 2 6 0( a )

- 2 0 - 1 0 0 1 0 2 0- 3

- 2

- 1

0

1

2

3

f = 2 1 9 . 4 T H z ( - 1 7 . 7 T H z ) , G = 1 . 0 0 0 4 f = 2 2 5 . 4 T H z ( - 1 1 . 7 T H z ) , G = 1 . 0 0 0 6Ph

ase [

10-4 ra

d]

R e l a t i v e F r e q u e n c y [ T H z ]

2 2 0 2 3 0 2 4 0 2 5 0 2 6 0( b )

A b s o l u t e F r e q u e n c y [ T H z ]

Figure 4.8: Calculated amplification (a) and induced phase modulation (b) of the signal on the outputof the summator after amplification by a single filter.

H(Z) =B

1 +AZ−1(4.15)

and in the time space:

ESE,out(m, t) = BESE,in(m, t) + ZSE(m, t− dt)ZSE(m, t) = A(m)ESE,out(m, t) (4.16)

The coefficients A(m) and B are equal to the coefficients for gain filters. Magnitude and phaseresponses of the two SE filters are presented in fig. 4.11.

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(a)

1 5 0 2 0 0 2 5 0 3 0 01 . 0 0 0 0

1 . 0 0 0 2

1 . 0 0 0 4

1 . 0 0 0 6

1 . 0 0 0 8

1 . 0 0 1 0

1 . 0 0 1 2

Magn

itude

Res

pons

e

F r e q u e n c y [ T H z ]

d t = 2 0 f s d t = 5 f s

(b)

1 5 0 2 0 0 2 5 0 3 0 0- 0 . 0 0 0 3

- 0 . 0 0 0 2

- 0 . 0 0 0 1

0 . 0 0 0 0

0 . 0 0 0 1

0 . 0 0 0 2

0 . 0 0 0 3

Phas

e Res

pons

e [rad

]

F r e q u e n c y [ T H z ]

d t = 2 0 f s d t = 5 f s

Figure 4.9: Calculated amplification (a) and induced phase modulation (b) of the signal on the outputof the summator after amplification by a single filter for different calculation time steps (20 fs and 5 fs).

(a)

1 2 7 5 1 3 0 0 1 3 2 5 1 3 5 0 1 3 7 51 . 0 0 0 0

1 . 0 0 0 5

1 . 0 0 1 0

1 . 0 0 1 5

Mater

ial G

ain

W a v e l e n g t h [ n m ]

S i n g l e Q D G S e n s e m b l e

(b)

1 2 7 5 1 3 0 0 1 3 2 5 1 3 5 0 1 3 7 5- 0 . 0 0 0 8- 0 . 0 0 0 6- 0 . 0 0 0 4- 0 . 0 0 0 20 . 0 0 0 00 . 0 0 0 20 . 0 0 0 40 . 0 0 0 60 . 0 0 0 8

Ph

ase M

odula

tion [

rad]

W a v e l e n g t h [ n m ]

S i n g l e Q D G S e n s e m b l e

Figure 4.10: Calculated magnitude (a) and phase (b) responses of a single gain filter and multiple filterswithin inhomogeneous distribution. The total amplification is calculated as a summation over all filters.

- 2 0 - 1 0 0 1 0 2 00 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0 f = 2 1 9 . 4 T H z ( - 1 7 . 7 T H z ) f = 2 2 5 . 4 T H z ( - 1 1 . 7 T H z )

Magn

itude

R e l a t i v e F r e q u e n c y [ T H z ]

2 2 0 2 3 0 2 4 0 2 5 0 2 6 0( a )

A b s o l u t e F r e q u e n c y [ T H z ]

- 2 0 - 1 0 0 1 0 2 0- 1 . 5

- 1 . 0

- 0 . 5

0 . 0

0 . 5

1 . 0

1 . 5 f = 2 1 9 . 4 T H z ( - 1 7 . 7 T H z ) f = 2 2 5 . 4 T H z ( - 1 1 . 7 T H z )

A b s o l u t e F r e q u e n c y [ T H z ]

Phas

e [rad

]

R e l a t i v e F r e q u e n c y [ T H z ]

2 2 0 2 3 0 2 4 0 2 5 0 2 6 0( b )

Figure 4.11: Calculated magnitude (a) and phase (b) responses of SE digital filters with transitionfrequencies of 219.4 THz and 225.4 THz (relative frequencies in the model are -17.7 THz and -11.7 THz).

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Chapter 5

Modeling of QD-SOA static behavior

5.1 Amplified spontaneous emission

Calculation and measurement of the amplified spontaneous emission are one of the basic characteristicof QD-SOA and describe the unsaturated behavior of the QD-SOA. This allows to determine the valuesof the inhomogeneous broadening as well as the spontaneous recombinations. Figure 5.1 presents thecalculated and the measured ASE for different injection currents.

It is visible that by increasing the current from 50 mA to 250 mA, we receive an increase of GS ASE,and the wavelength of maximal ASE intensity is shifted to shorter region (the blue shift) by increasingthe current. For the further increasing of the injected current the peak ASE wavelength is shifted intothe longer direction (the red shift) and peak intensity is decreased.

The blue shift occurs due to different energy separations between QW and QD levels. Small QDswith higher transition energies (and shorter wavelengths) have smaller capturing and escaping times,compared to large ones. So carriers, that are captured by small dots, have a good probability to escapeback to the QW, but carriers captured by the large QDs will stay there for a longer time. This leadsto higher carrier concentrations in the QDs with longer wavelengths. For further increase the injectionscurrents, there are more carriers in the active regions and they fulfill all QDs more uniformly. This effectis not important for operational currents of QD-SOA (above 250 mA), so it is not included in the model.

At the injected current value of 250 mA the GS is completely filled with carriers and has a maximallinear gain (18 dB) and ASE intensity (-29 dB/nm).

The red shift was not described by publications as we know, so we suppose that the reason isthe increase in the QD-SOA temperature, which leads to the change of material strain for QDs andtherefore changes their transition energies. This effect depends on the size of the QDs, so energies arechanging unevenly, which leads to increasing energy separations between QDs. Thus the inhomogeneousbroadening is increasing and therefore decrease the linear gain. The other possible reason is a bandgaprenormalization, equal to bulk [78] and QW structures. High carrier density in QDs change electromagneticfield in the crystal and thus decreases energy levels .

An approximation was used to fit the experimental results. The red shifted central GS transitionenergy EGS and the changed energy separation between QD groups ∆EGS in dependence of the injectedcurrent IInj is given by:

EGS(IInj) = EGS,01

1 + kRSGS(IInj − 250mA)(5.1)

∆EGS(IInj) = ∆EGS,0(1 + kRS∆E(IInj − 250mA)

)where EGS,0 and ∆EGS,0 are GS transition energy and separation between QD groups for IInj=250 mA,kRSGS and kRS∆E are linear coefficients.

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1 2 0 0 1 2 5 0 1 3 0 0 1 3 5 0- 6 4- 6 0- 5 6- 5 2- 4 8- 4 4- 4 0- 3 6- 3 2- 2 8- 2 4- 2 0

G SE S

E x p e r i m e n t : T h e o r y : 5 0 0 m A 5 0 0 m A 2 5 0 m A 2 5 0 m A 5 0 m A

Inten

sity [

dBm/

0.5nm

]

W a v e l e n g t h [ n m ]

Figure 5.1: Calculated and measured amplified spontaneous emission for different injection currents.

0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 1 . 4 1 . 6 1 . 8 2 . 0- 4 5- 4 0- 3 5- 3 0- 2 5- 2 0- 1 5- 1 0

- 50

To

tal op

tical

powe

r [dB

m]

Q D - S O A l e n g t h [ m m ]

F o r w a r d B a c k w a r d

Figure 5.2: Calculated distribution of total optical power of ASE for forward and backward propagationsfor injected current 250 mA.

5.1.1 Carriers dynamics

As the optical power in the QD-SOA is in the linear regime, as presented in Fig. 5.2, the main recombinationmechanism is the spontaneous recombination, as visible in Fig. 5.3. This leads to a flat profile of carrierdensities over QD-SOA length and over the inhomogeneous distribution of QDs, as presented in Fig. 5.4.

Carrier dynamics for the intraband transition are presented in Fig. 5.5. Both excitation and escapingprocesses are very important and therefore should be included in the modeling.

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2 4 6 8 1 0 1 2 1 4 1 60 . 0 0 0 50 . 0 0 1 00 . 0 0 1 50 . 0 0 2 00 . 0 0 2 50 . 0 0 3 00 . 0 0 3 50 . 0 0 4 00 . 0 0 4 5

0 . 1 0

0 . 1 5

0 . 2 0

Reco

mbina

tion r

ates [

1010

s-1 ]

N u m b e r o f Q D - g r o u p

S p o n t a n e o u s r e c o m b i n a t i o n : G S E S

S t i m u l a t e d r e c o m b i n a t i o n : G S E S

Figure 5.3: Calculated rates of stimulated and spontaneous carriers recombinations for intradots levelsover the inhomogeneous distribution of QDs for the end segment of QD-SOA (injected current 250 mA).

0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 1 . 4 1 . 6 1 . 8 2 . 00

1

2

3

4

5

6( a )

2D ca

rriers

dens

ities [

1015

m-2 ]

Q D - S O A l e n g t h [ m m ]

n _ Q W _ e n _ Q W _ h n _ T o t a l _ e n _ T o t a l _ h

2 4 6 8 1 0 1 2 1 4 1 60 . 6 2 5

0 . 6 5 0

0 . 9 0 0

0 . 9 2 5

0 . 9 5 0

0 . 9 7 5

1 . 0 0 0( b )

Occu

patio

n Pro

babil

ity

N u m b e r o f Q D - g r o u p

E S _ e E S _ h G S _ e G S _ h

Figure 5.4: Calculated carrier distributions for electrons (e) and holes (h) for injected current 250 mA.2D QW and total carrier densities over QD-SOA length (a), and occupation probabilities over theinhomogeneous distribution of QDs for the end segment of QD-SOA (b).

2 4 6 8 1 0 1 2 1 4 1 60 . 0

0 . 2

0 . 4

0 . 6( a )

Carri

ers dy

nami

cs [1

010 s-1 ]

N u m b e r o f Q D - g r o u p

Q W - E S _ e Q W - E S _ h E S - Q W _ e E S - Q W _ h

2 4 6 8 1 0 1 2 1 4 1 60

2

4

6

8( b )

Carri

ers dy

nami

cs [1

010 s-1 ]

N u m b e r o f Q D - g r o u p

Q W - G S _ e Q W - G S _ h G S - Q W _ e G S - Q W _ h E S - G S _ e G S - E S _ e E S - G S _ h G S - E S _ h

Figure 5.5: Calculated carrier dynamics for electrons (e) and holes (h) over the inhomogeneousdistribution of QDs for the end segment of QD-SOA for ES (a) and GS (b) for injected current 250mA. Transitions between QW, ES and GS are presented.

35

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5.2 Continuous wave

The CW signal propagation allows to determinate the main carrier dynamics. The experiments andsimulations show QD-SOA behavior under static saturation.

Figure 5.6 shows the net gain as a function of the output optical power for different injection currentsfor 1310 nm CW signals.

Selected wavelength of 1310 nm corresponds to the maximum gain for the current 250 mA, while theunsaturated gain at this wavelength for 500 mA is smaller due to the red shift and it is required to correctthe signal wavelengths for different currents to receive higher linear gain.

- 4 - 2 0 2 4 6 8 1 0 1 2 1 4 1 6 1 89

1 01 11 21 31 41 51 61 71 8

Gain

[dB]

O u t p u t P o w e r [ d B m ]

T h e o r y : E x p e r i m e n t : 5 0 0 m A 5 0 0 m A 2 5 0 m A 2 5 0 m A

Figure 5.6: Calculated and measured static gain saturation for different injected currents for 1310 nmCW signals with different optical powers. For high current we observe smaller unsaturated linear gain(due to the red shift) and higher saturation power (due to higher amount of carriers in the active region).

- 2 0 2 4 6 8 1 0 1 2 1 4 1 60 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0( a )

Occu

patio

n Prob

abilit

y

O u t p u t P o w e r [ d B m ]

f _ G S _ e f _ G S _ h f _ E S _ e f _ E S _ h

- 2 0 2 4 6 8 1 0 1 2 1 4 1 60

2

4

6( b )

2D C

arrier

Den

sity [

1015

m-2 ]

O u t p u t P o w e r [ d B m ]

n _ Q W _ e n _ Q W _ h n _ T o t a l _ e n _ T o t a l _ h

Figure 5.7: Calculated carrier distribution in the last segment of the QD-SOA for different optical powersand the injected current of 250 mA. Occupation probabilities (a) are calculated for the central QD in theinhomogeneous distribution with transition wavelength equal to the input signal. 2D carrier densities arepresented for QW (n QW e(h)) as well as the total carrier density (n Total e(h)) per layer (b).

The QD-SOA gain is almost constant for output powers of up to 0 dBm for the current 250 mAand up to 13 dBm for the current 500 mA. A small decrease in this nearly linear regime occurs due toa decrease of the carrier density of the ES and the QW, while the GS still has enough carriers. Thisbehavior leads to a higher saturation power, compared to bulk or QW-SOAs. For higher optical powerwe see a dramatic decrease of the gain since the device does not have enough carriers to fill the GS. Thevalue of the saturation power is mainly determinated by the injected current and the scattering ratesinto the GS. Figure 5.7 shows carrier distributions for different optical powers. As visible for the smalloptical power QD-SOA is unsaturated and carrier densities are equal to the unsaturated QD-SOA.

Figure 5.8 presents the calculated distribution of the total optical power for forward and backwardpropagations and material gain for injected current of 250 mA and input optical power of 5 dBm. As

36

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0 . 0 0 0 . 2 5 0 . 5 0 0 . 7 5 1 . 0 0 1 . 2 5 1 . 5 0 1 . 7 5 2 . 0 0- 6 0- 5 0- 4 0- 3 0- 2 0- 1 0

01 02 0( a )

F o r w a r d R e v e r s e

Optic

al Po

wer [

dBm]

Q D - S O A L e n g t h [ m m ]0 . 0 0 0 . 2 5 0 . 5 0 0 . 7 5 1 . 0 0 1 . 2 5 1 . 5 0 1 . 7 5 2 . 0 00123456789

1 0( b )

Gain

[dB]

Q D - S O A L e n g t h [ m m ]

Figure 5.8: Calculated distribution of total optical power for forward and backward propagations (a) andaccumulaated material gain (b) for injected current of 250 mA and input optical power of 5 dBm.

0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 1 . 4 1 . 6 1 . 8 2 . 00

1

2

3

4

5

6( a )

2D ca

rriers

dens

ities [

1015

m-2 ]

Q D - S O A l e n g t h [ m m ]

n _ Q W _ e n _ Q W _ h n _ T o t a l _ e n _ T o t a l _ h

2 4 6 8 1 0 1 2 1 4 1 60 . 00 . 10 . 20 . 30 . 40 . 50 . 60 . 70 . 80 . 91 . 0( b )

Occu

patio

n Pro

babil

ity

N u m b e r o f Q D - g r o u p

f _ E S _ e _ ( 0 ) f _ E S _ e _ ( L ) f _ E S _ h _ ( 0 ) f _ E S _ h _ ( L ) f _ G S _ e _ ( 0 ) f _ G S _ e _ ( L ) f _ G S _ h _ ( 0 ) f _ G S _ h _ ( L )

Figure 5.9: Calculated carrier densities over QD-SOA length (a) and over the inhomogeneous distributionof QDs for the first (labeled as “0”) and end (labeled as “L”) segments of QD-SOA (b).

visible, the optical signal power is significantly higher than ASE, so the noise can be neglected for thiskind of simulations. The gain distribution can be divided into two main sections - linear gain sectionin the beginning and saturated section in the middle and output segments of the QD-SOA. Gain valuesare determined by the carrier densities as presented in Fig. 5.9. For the first segments we have smalloptical power and carrier densities are equal to the unsaturated QD-SOA, while significant saturationand spectral hole burning can be observed on the output segment. This is an ordinary behavior for aSOA.

Figures 5.10 and 5.11 illustrate values of carrier transitions. Compared to ASE simulations (fig. 5.4and fig 5.5)), we observe higher capturing and relaxations for carriers, as less carriers in QDs means morefree space for new ones.

As visible, for high optical power we observe absorption by ES due to negligible carrier density onthis energy level. Anyway that does not change the optical power significantly as optical frequency is farfrom ES resonance and this absorption is created only by long tails of the homogeneous broadening.

High stimulated recombination reduces carrier densities on the GS, thus increases intradot relaxationrate (Fig. 5.11) and reduce ES carrier density (Fig. 5.9). Decreasing the ES density leads to the highercapturing rates and smaller escaping (Fig. 5.11) thus decreases the carrier density for the QW.

37

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2 4 6 8 1 0 1 2 1 4 1 6- 0 . 2

0 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0

Carri

ers dy

nami

cs [1

010 s-1 ]

N u m b e r o f Q D - g r o u p

s t i m _ G S s t i m _ E S s p o n t _ G S s p o n t _ E S

Figure 5.10: Calculated rates of stimulated (labeled as “stim”) and spontaneous (labeled as “spont”)carrier recombinations for intradots levels over the inhomogeneous distribution of QDs for the end segmentof QD-SOA for input optical power of 5 dBm and injected current of 250 mA.

2 4 6 8 1 0 1 2 1 4 1 60 . 00 . 10 . 20 . 30 . 40 . 50 . 60 . 7( a )

Carri

ers dy

nami

cs [1

010 s-1 ]

N u m b e r o f Q D - g r o u p

Q W - E S _ e W Q - E S _ h E S - Q W _ e E S - W Q _ h

2 4 6 8 1 0 1 2 1 4 1 60 . 0 00 . 0 50 . 1 00 . 1 50 . 2 0

51 01 52 02 5

Carri

ers dy

nami

cs [1

010 s-1 ]

N u m b e r o f Q D - g r o u p

Q W - G S _ e Q W - G S _ h G S - Q W _ e G S - Q W _ h E S - G S _ e G S - E S _ e E S - G S _ h G S - E S _ h

( b )

Figure 5.11: Calculated carrier dynamics for electrons (e) and holes (h) over the inhomogeneousdistribution of QDs for the end segment of QD-SOA for ES (a) and GS (b) for input optical power5 dBm and injected current 250 mA. Transitions between QW, ES and GS are presented.

38

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Chapter 6

Modeling of QD-SOA dynamicbehavior

6.1 Pump-probe behavior

In this section we will discuss the pump-probe behavior of the QD-SOA for pulse sequences. This behaviorhas been well investigated and represented by conventional models [79–84]. The core of these simulationsand experiments is to determine a QD-SOA modulation for optical pump pulses with sufficient power.These pulses modulate carrier densities and induce additional nonlinear effects that are measured bythe probe pulses with different wavelengths. Utilization of short probe pulses allows to receive a bettertime resolution. Experimental probe signals were pulse sequences with different delays around the pumppulses, while for simulations we have directly calculated the QD-SOA gain and induced phase shift forprobe wavelengths. The optical power of the probe signals should be negligible to prevent additionalmodulation. These types of experiments and simulations describe the dynamic behavior of the QD-SOA.

6.1.1 Cross-Gain Modulation

The presented experiments were carried out for pulses with 1.5 ps full width at half maximum (FWHM)pump pulses with hyperbolic secant squared shape and the central wavelength at 1310 nm. Probe pulsesin the experiments have hyperbolic secant squared shape and FWHM duration of 1.3 ps.

0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 056789

1 01 11 21 3( a )

Gain

[dB]

T i m e [ p s ]

E x p e r i m e n t T h e o r y

0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 056789

1 01 11 21 3

Gain

[dB]

T i m e [ p s ]

E x p e r i m e n t T h e o r y

( b )

Figure 6.1: Measured and calculated pump-probe dynamics for pulses with 10 GHz repetition rate forinjected current 250 mA (a) and 500 mA (b) for average input pump power of 5 dBm and 1310 nm pumpand 1302 nm probe wavelengths.

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0 5 1 0 1 5 2 0 2 5 3 0 3 56

7

8

9

1 0

1 1( a )

Gain

[dB]

T i m e [ p s ]

T h e o r y E x p e r i m e n t

0 5 1 0 1 5 2 0 2 5 3 0 3 5

8

9

1 0

1 1( b )

Gain

[dB]

T i m e [ p s ]

T h e o r y E x p e r i m e n t

Figure 6.2: Measured and calculated pump-probe dynamics for pulses with 40 GHz repetition rate forinjected current 250 mA (a) and 500 mA (b) for average input pump power 5 dBm and 1310 nm pumpand 1302 nm probe wavelengths.

0 1 2 3 4 5 60 . 00 . 10 . 20 . 30 . 40 . 50 . 60 . 70 . 80 . 9( a )

Optic

al Po

wer [

W]

T i m e [ p s ]

P _ o u t P _ i n

0 2 0 4 0 6 0 8 0 1 0 0- 8- 7- 6- 5- 4- 3- 2- 10( b )

Gain

supp

ressio

n [dB

]

T i m e [ p s ]

T o t a l G a i n S u p p r e s s i o n T P A S u p p r e s s i o n M a t e r i a l G a i n S u p p r e s s i o n

Figure 6.3: Calculated input (labeled as “in”) and output (labeled as “out”) pump optical powers (a)and relative net gain saturation due to TPA and material gain (b). Simulations were provide for 10 GHzrepetition rate with average input optical power 5 dBm and an injected current of 250 mA.

Figures 6.1 and 6.2 present calculated and measured gain saturations for 10 GHz and 40 GHz pumppulses for 250 mA and 500 mA injected currents. Pump wavelength of 1310 nm was selected for themaximum amplification for 250 mA current. Probe wavelength was 1302 nm.

As visible, for 10 GHz pulses for 500 mA we have lower gain at the moment before the pulses as theunsaturated gain value for probe wavelength is lower due to the red shift. Lower gain saturation for highcurrent has the following reasons:- lower initial gain leads to smaller stimulated recombination and lower carrier depletion.- for high current we have more carriers in the active region that provides faster GS recovery during thepulse propagation and decreases the carrier density modulation for GS.

For 40 GHz pulses with the same average energy we observe significantly smaller gain pulsations asthe input energy per pulse is 4 times smaller. This reduces stimulated recombination per pulse and TPA.For a higher pulse frequency results are more similar to the CW measurements, as optical power is moremonotonous.

Figure 6.3a presents calculated input and output optical powers. As visible, the main amplificationoccurs on the first edge of the pulses, while the amplification of the second edge is significantly saturated.This is the usual behavior for the SOA and leads to the time shift of the peak for the output pulses,compared to the input. Another important aspect is the significant maximal optical power, that canexceed 1 W for short pulses and thus makes TPA very important.

40

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0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 00 . 3

0 . 4

0 . 5

0 . 6

0 . 7

0 . 8

0 . 9

1 . 0( a )

Occu

patio

n Pro

babil

ity

T i m e [ p s ]

f _ G S _ e ( 1 3 0 2 n m ) f _ G S _ e ( 1 3 0 8 n m ) f _ G S _ h ( 1 3 0 2 n m ) f _ G S _ h ( 1 3 0 8 n m )

0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 00 . 00 . 10 . 20 . 30 . 40 . 50 . 60 . 70 . 80 . 91 . 0( b )

Occu

patio

n Pro

babil

ity

T i m e [ p s ]

f _ E S _ e ( 1 3 0 2 n m ) f _ E S _ e ( 1 3 0 8 n m ) f _ E S _ h ( 1 3 0 2 n m ) f _ E S _ h ( 1 3 0 8 n m )

0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 00

1

2

3

4

5( c )

2D C

arrier

Den

sites

[1015

m-2 ]

T i m e [ p s ]

n _ Q W _ e n _ Q W _ h n _ T o t a l _ e n _ T o t a l _ h

Figure 6.4: Calculated intradot carriers densities for QDs with resonance frequencies near pump andprobe signals (a, b) and 2D carrier dencities per QD layer QW (c) in the active region.

0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0

0 . 0 0 0

0 . 0 2 5

0 . 0 5 0

0 . 2 80 . 2 90 . 3 00 . 3 1( a )

Stim

ulated

Rec

ombin

ation

[ps-1 ]

T i m e [ p s ]

s t i m _ G S ( 1 3 0 2 n m ) s t i m _ G S ( 1 3 0 8 n m ) s t i m _ E S ( 1 3 0 2 n m ) s t i m _ E S ( 1 3 0 8 n m )

0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 00 . 0 0 0 0

0 . 0 0 0 5

0 . 0 0 1 0

0 . 0 0 1 5

0 . 0 0 2 0

0 . 0 0 2 5 ( b )

Spon

taneo

us R

ecom

binati

on [p

s-1 ]

T i m e [ p s ]

s p o n t _ G S ( 1 3 0 2 n m ) s p o n t _ G S ( 1 3 0 8 n m ) s p o n t _ E S ( 1 3 0 2 n m ) s p o n t _ E S ( 1 3 0 9 n m )

Figure 6.5: Calculated stimulated (a) and spontaneous (b) recombinations for the intradot energy levels inthe last segment of QD-SOA. Recombinations are presented for QDs with different transition frequenciesnear pump and probe signals.

Figure 6.3b shows an influence of the TPA and the material gain on the net gain saturation. Asvisible, the gain recovery consists of two parts - fast and slow. Fast gain recovery takes place due to theinstantaneous TPA and the intradot transitions ES→GS. TPA creates additional losses with the sameshape as optical pulses.

Calculated carrier dynamics are presented in Fig. 6.4. Optical pulses induce stimulated recombinationand reduce GS carrier density. After this we can see high ES→GS relaxation and ES is working as asmall carrier reservoir. This creates fast material gain recovery and reduces ES carrier density (speciallyfor electrons). As far as ES in the conduction band becomes empty the GS gain recovery is determinatedby the capturing processes QW→ES that are slower. Calculated carriers transitions are presented inFig. 6.5−6.8.

As visible, for short optical pulses we observe a very high stimulated recombination rate (0.3 ps−1)that significantly decrease carrier lifetime and dephasing time specially for electrons, as they have smallercarrier density. As we loose ≈70% of the electrons per picosecond (at the time moment around 26 ps onplots). This leads to an increase of the homogeneous broadening. This effect shoud be investigated inmore details and placed in the model to receive more detailed simulations.

Figure 6.9 presents the modulation of the QD carrier densities. As visible, we obtain significantspectral hole (better visible for electrons) for QDs with transition wavelength around the signal. Broadeningof this hole depends on the homogeneous broadening. Modulation for the other QDs is lower and dependsmore on the QW carrier density pulsations.

41

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0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 00 . 0 0 0 0 00 . 0 0 0 0 10 . 0 0 0 0 20 . 0 0 0 0 30 . 0 0 0 0 40 . 0 0 2 0

0 . 0 0 2 5

0 . 0 0 3 0

0 . 0 0 3 5( a )

( a )

Carri

ers Tr

ansit

ions [

ps-1 ]

T i m e [ p s ]

Q W _ E S _ e ( 1 3 0 2 n m ) Q W _ E S _ e ( 1 3 0 8 n m ) E S _ Q W _ e ( 1 3 0 2 n m ) E S _ Q W _ e ( 1 3 0 8 n m )

0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 00 . 0 0 0

0 . 0 0 2

0 . 0 0 4

0 . 0 0 6

0 . 0 0 8

0 . 0 1 0

0 . 0 1 2( b )

Carri

ers Tr

ansit

ions [

ps-1 ]

T i m e [ p s ]

Q W _ E S _ h ( 1 3 0 2 n m ) Q W _ E S _ h ( 1 3 0 8 n m ) E S _ Q W _ h ( 1 3 0 2 n m ) E S _ Q W _ h ( 1 3 0 8 n m )

Figure 6.6: Calculated carrier transitions between QW and ES for electrons (a) and holes (b) in the lastsegment of the QD-SOA for QDs with different transition wavelengths. Input optical power is 5 dBmand injected current is 250 mA.

0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 00 . 0 0 0 0 0 00 . 0 0 0 0 0 20 . 0 0 0 0 0 40 . 0 0 0 0 0 60 . 0 0 0 0 0 80 . 0 0 0 0 1 0

0 . 0 0 0 5

0 . 0 0 1 0

0 . 0 0 1 5( a )

Carri

ers Tr

ansit

ions [

ps-1 ]

T i m e [ p s ]

Q W _ G S _ e ( 1 3 0 2 n m ) Q W _ G S _ e ( 1 3 0 8 n m ) G S _ Q W _ e ( 1 3 0 2 n m ) G S _ Q W _ e ( 1 3 0 8 n m )

0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 00 . 0 0 1

0 . 0 0 2

0 . 0 0 3

0 . 0 0 4

0 . 0 0 5

0 . 0 0 6( b )

Carri

ers Tr

ansit

ions [

ps-1 ]

T i m e [ p s ]

Q W _ G S _ h ( 1 3 0 2 n m ) Q W _ G S _ h ( 1 3 0 8 n m ) G S _ Q W _ h ( 1 3 0 2 n m ) G S _ Q W _ h ( 1 3 0 8 n m )

Figure 6.7: Calculated carrier transitions between QW and GS for electrons (a) and holes (b) in the lastsegment of the QD-SOA for QDs with different transition wavelengths. Input optical power is 5 dBmand injected current is 250 mA.

6.1.2 Cross-Phase Modulation

Cross-phase modulation (XPM) allows to measure the phase modulation of the probe signal in relationto the pump signal. Multiple experiments have indicated that optical phase modulation has significantlyslower dynamics as compared to the optical gain [85–95]. Other theoretical model can reproduce thisphase behavior, but they do not include the frequency dependence of the phase modulation.

Pump-probe (PP) simulations and experiments were provided for a 10 GHz pulse sequence (FWHM 1.5 ps)with an average optical power of 3.5 dBm, injected current 350 mA and a central wavelength 1310 nm.Probe signals have wavelengths of 1301 nm and 1319 nm for positive and negative detunings, respectively.Experimental probe signals were 10 GHz pules sequences (FWHM 1.3 ps).

There are two alternatives to determinate the phase modulation - as a difference between output φoutand input φin phases (that is used in the presented results), or between input and output phases:

∆φ = φout − φin (6.1)

∆φ = φin − φout

Both of them are widely used in the literature and this may provoke a lot of misunderstandings.Figures 6.10 and 6.11 show measured and calculated XGM and XPM for both probe wavelengths. We

can see that both TPA and FCA create the same additional phase shift for both detuning frequencies butthe only difference is the phase modulation due to the material gain. Refractive index modulation due

42

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0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 00 . 0 00 . 0 10 . 0 20 . 0 30 . 0 40 . 0 50 . 0 60 . 0 70 . 0 80 . 0 90 . 1 0( a )

Carri

ers Tr

ansit

ions [

ps-1 ]

T i m e [ p s ]

E S _ G S _ e ( 1 3 0 2 n m ) E S _ G S _ e ( 1 3 0 8 n m ) G S _ E S _ e ( 1 3 0 2 n m ) G S _ E S _ e ( 1 3 0 8 n m )

0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 00 . 0

0 . 1

0 . 2

0 . 3

0 . 4

0 . 5( b )

Carri

ers Tr

ansit

ions [

ps-1 ]

T i m e [ p s ]

E S _ G S _ h ( 1 3 0 2 n m ) E S _ G S _ h ( 1 3 0 8 n m ) G S _ E S _ h ( 1 3 0 2 n m ) G S _ E S _ h ( 1 3 0 8 n m )

Figure 6.8: Calculated intradot carrier transitions for electrons (a) and holes (b) in the last segment ofthe QD-SOA for QDs with different transition wavelengths. Input optical power 5 dBm and injectedcurrent 250 mA.

2 4 6 8 1 0 1 2 1 4 1 60 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0

Carri

ers D

ensit

y [10

14 m

-2 ]

N u m b e r o f Q D - g r o u p

f _ G S _ e _ m i n f _ G S _ e _ m a x f _ G S _ h _ m i n f _ G S _ h _ m a x

Figure 6.9: Calculated minimum and maximum carrier densities for different QDs in the inhomogeneousbroadening.

to GS material gain has almost the same value for both detunings, but phase modulation has differentsigns that is negative for the shorter wavelength and positive for the longer wevelength. A small increasefor a gain induced phase modulation for the shorter wavelength is created by the influence of the ES. Ahigher value of the total phase modulation for the positive detuning is specific for the QD structures andwas previously discovered by this theoretical work and confirmed later by the experiments.

Figure 6.12 shows calculated carrier distributions for a QD group with a reference frequency equal tothe pump signal frequency and for QW and total 2D carriers densities per layer.

In comparison to the phase dynamics (Fig. 6.10 and 6.11) we can see, that FCA created by carriersin the QDs has the same shape as the carrier density modulation in QDs, while FCA in the QW isproportional to the QW carrier densities. QW FCA is the slowest XPM process as it is created by QWcarrier modulation. It has the maximum around 30 ps after the pulses coming and decreasing speed isthe same as QW carrier recovering rate.

TPA by its own does not create additional phase modulation, but the associated alpha-factor and Kerreffect do. These processes retrace the form of the optical pulses and create ultrafast phase modulation.

It is visible, that for longer probe wavelengths we obtain higher XPM for near 15 ps after the pulse,and almost the same phase recovery for the following time.

43

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0 2 0 4 0 6 0 8 0 1 0 0 1 2 0- 7- 6- 5- 4- 3- 2- 10( a )

Relat

ive G

ain [d

B]

T i m e [ p s ]

E x p e r i m e n tT h e o r i e :

N e t M o d u l a t i o n G a i n T P A

0 2 0 4 0 6 0 8 0 1 0 0 1 2 0- 0 . 2

0 . 0

0 . 2

0 . 4

0 . 6( b )

Relat

ive Ph

ase [

rad]

T i m e [ p s ]

E x p e r i m e n tT h e o r i e :

N e t M o d u l a t i o n G a i n T P A Q D - F C A Q W - F C A

Figure 6.10: Influence of the material gain, FCA and TPA on the amplification and the phase modulationof the probe signal (λ=1301 nm).

0 2 0 4 0 6 0 8 0 1 0 0 1 2 0- 8- 7- 6- 5- 4- 3- 2- 10( a )

Relat

ive G

ain [d

B]

T i m e [ p s ]

E x p e r i m e n tT h e o r i e :

N e t M o d u l a t i o n G a i n T P A

0 2 0 4 0 6 0 8 0 1 0 0 1 2 00 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0( b )Re

lative

Phas

e [rad

]

T i m e [ p s ]

E x p e r i m e n tT h e o r i e :

N e t M o d u l a t i o n G a i n T P A Q W - F C A Q D - F C A

Figure 6.11: Influence of the material gain, FCA and TPA on the amplification and the phase modulationof the probe signal (λ=1319 nm).

0 2 0 4 0 6 0 8 0 1 0 0 1 2 00 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0( a )

Occu

patio

n Pro

babil

ity

T i m e [ p s ]

G S - e l e c t r o n s G S - h o l e s E S - e l e c t r o n s E S - h o l e s

0 2 0 4 0 6 0 8 0 1 0 0 1 2 0

1

2

3

4

5( b )

Carri

er De

nsity

[1015

1/m2 ]

T i m e [ p s ]

T o t a l - h o l e s T o t a l - e l e c t r o n s Q W - h o l e s Q W - e l e c t r o n s

Figure 6.12: Carriers dynamics in the last SOA segment for intradot levels (GS and ES), QW and totalcarrier density for PP simulations. Occupation probabilities are for a QD-group with the resonancewavelength equal to the pump ((λ=1310 nm)).

44

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6.2 Frequency chirp

The chirp represents a shift of the signal frequency due to the nonlinear processes while propagatingthrough the active medium of a semiconductor. This effect can be very important in telecommunicationsystems as it produces additional distortion for WDM or phase-modulated signals. The physical reason ofit is the modulation of the refractive index of the waveguide, that leads to an additional phase modulationof the signal.

Previous publications have pointed a very low alpha-factor for QD-SOA and even negative values canbe obtained due to the discrete energy levels of QDs.

In this section we will present theoretical calculations of the chirp for the input pulses with differentcarrier wavelengths and optical powers.

The value of the chirp can be calculated by using the following relation:

dν = − 1

dt(6.2)

where φ is the optical phase.All effects that create additional phase shift are included in the presented model and are described in

the previous section. Here we will briefly present the calculated results.Figure 6.13 presents calculated output optical power, phase shift and chirp for propagation of 10 GHz

optical pulses (1.5 ps FWHM) with different wavelengths and input optical powers for injected currentof 250 mA. All processes were included in this simulations except ASE to receive more “clean” results.

For small average input powers (-20 dBm) the output power greatly depends on the wavelengths, as thegain spectrum depends on the inhomogeneous broadening of QDs. The detunings are symmetric aroundthe maximum amplification wavelength (1310 nm), so amplification for wavelengths in pair (1300 nmand 1320 nm, 1290 nm and 1330 nm) are almost equal. A small decrease for shorter wavelengths comesfrom additional absorption from ES. Induced phase modulations are relatively small and almost zero for1310 nm as we do not have the refractive index modulation from the gain for this wavelength in thecenter of the inhomogeneous broadening.

Calculated chirp is the function of induced phase shift. Created chirps for symmetrical wavelengthshave nearly the same values but different signs. Asymmetry is created by negligible influences of FCAand TPA processes as well as carriers on ES. For 1310 nm pulses we observe positive chirp for the firstfront of the pulses and negative for the second front. This effects comes from the fact, that signal havemultiple spectral components that have different phase modulations. Anyway the absolute value of thechirp is less than 2 GHz that is significantly smaller as for bulk or QW-structures.

For the high average input optical power (0 dBm) we observe smaller frequency dependence of the netgain due to the saturation of QD-SOA and the value of the gain depends mainly on the number of carriersin the active zone, not on the value of the unsaturated gain. Another reason is TPA that creates higherabsorption for high optical power and thereby equalize output optical powers for different wavelengths.

Phase modulations are significantly higher for high optical power and FCA becomes the dominatingprocess. As FCA has negligible frequency dependence, we observe significant chirp of the same signs forall signal wavelengths.

Figure 6.14 presents normalized output optical powers and their corresponding spectra. For smalloptical powers induced distortions are negligible and both pulse shapes and spectra are symmetric. Forhigh optical power the first front of the pulses is more strongly amplified, this leads to a shift of thecentral position of the pulse and increases its broadening. The spectra become narrowe due to the shiftof short-wave components. The same behavior was experimentally observed for a bulk SOA [96].

45

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0 1 2 3 4 5 60 . 0 0

0 . 0 1

0 . 0 2

0 . 0 3

0 . 0 4

Outpu

t Opti

cal P

ower

[W]

T i m e [ p s ]

1 2 9 0 n m 1 3 0 0 n m 1 3 1 0 n m 1 3 2 0 n m 1 3 3 0 n m

0 1 2 3 4 5 6

- 0 . 1 0

- 0 . 0 5

0 . 0 0

0 . 0 5

0 . 1 0

Relat

ive Ph

ase S

hift [r

ad]

T i m e [ p s ]

1 2 9 0 n m 1 3 0 0 n m 1 3 1 0 n m 1 3 2 0 n m 1 3 3 0 n m

0 1 2 3 4 5 6- 1 2- 1 0- 8- 6- 4- 202468

1 01 2

Chirp

[GHz

]

T i m e [ p s ]

1 2 9 0 n m 1 3 0 0 n m 1 3 1 0 n m 1 3 2 0 n m 1 3 3 0 n m

0 1 2 3 4 5 60 . 00 . 10 . 20 . 30 . 40 . 50 . 60 . 70 . 8

Outpu

t Opti

cal P

ower

[W]

T i m e [ p s ]

1 2 9 0 n m 1 3 0 0 n m 1 3 1 0 n m 1 3 2 0 n m 1 3 3 0 n m

0 1 2 3 4 5 6

0 . 0

0 . 1

0 . 2

0 . 3

0 . 4

0 . 5Re

lative

Phas

e Shif

t [rad

]

T i m e [ p s ]

1 2 9 0 n m 1 3 0 0 n m 1 3 1 0 n m 1 3 2 0 n m 1 3 3 0 n m

0 1 2 3 4 5 6- 6 0- 5 0- 4 0- 3 0- 2 0- 1 0

01 0

Chirp

[GHz

]

T i m e [ p s ]

1 2 9 0 n m 1 3 0 0 n m 1 3 1 0 n m 1 3 2 0 n m 1 3 3 0 n m

Figure 6.13: Calculated output optical power, induced phase shift and chirp for optical pulses with theaverage input powers -20 dBm (above) and 0 dBm (below).

0 1 2 3 4 5 60 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0( a )

Norm

alize

d Pow

er

T i m e [ p s ]

P = - 2 0 d B m 1 2 9 0 1 3 0 0 1 3 1 0 1 3 2 0 1 3 3 0

P = 0 d B m 1 2 9 0 1 3 0 0 1 3 1 0 1 3 2 0 1 3 3 0

- 5 - 4 - 3 - 2 - 1 0 1 2 3 4 5- 5 0

- 4 0

- 3 0

- 2 0

- 1 0

0( b )

Norm

alize

d Inte

nsity

[dB]

R e l a t i v e W a v e l e n g t h [ n m ]

P = - 2 0 d B m 1 2 9 0 1 3 0 0 1 3 1 0 1 3 2 0 1 3 3 0

P = 0 d B m 1 2 9 0 1 3 0 0 1 3 1 0 1 3 2 0 1 3 3 0

Figure 6.14: Calculated normalized pulse shapes (a) and spectra (b) for the output optical signals withdifferent optical powers and wavelengths.

46

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6.3 Four-wave mixing

Four-wave mixing (FWM) is the nonlinear effect that occurs between two or more spectral componentsof the signal and leads to the generation of a new spectral components to the signal [6, 97–105].

The physical reason of this process is the modulation of the amplitude of the optical signal andcorresponding phase modulation, induced by a beating between different spectral components of thesignal.

For two CW input signals with frequencies f1 and f2 we will receive the new FWM components withfrequencies:

f3 =2f1 − f2

f4 =2f2 − f1

as schematically presented on fig. 6.15, where df = f2 − f1 is the detuning between input CW signals.Components with frequencies f5 and f6 were created as an interaction of the (f1 and f3) and (f2 and f4)respectively. This process is schematically presented in Fig. 6.15a.

As all nonlinear effects are included in the model, it is not required to add additional parameters forthe FWM simulations. The FWM products are received automatically for any type of input signal.

(a)

f

P

f1 f2

dfdf df

f3 f4

df

f5

df

f6

(b)

1 3 0 5 1 3 1 0 1 3 1 5 1 3 2 0 1 3 2 5 1 3 3 0 1 3 3 5- 8 0

- 6 0

- 4 0

- 2 0

0

2 0

f 6 f 5

f 4 f 3

f 2

P 1P 2f 1

Inten

sity [

dBm/

0.2nm

]

W a v e l e n g t h [ n m ]

Figure 6.15: Schematic representation of the of the spectrum with FWM products (a) and calculatedoutput signal with FWM products (b). Two CWs with frequencies f1 and f2 are input signal of theQD-SOA, and FWM-products with frequencies f3, f4, f5 and f6 were created during the nonlinearpropagation of the signal.

(a)

1 0 0 1 0 0 0- 8 0

- 7 0

- 6 0

- 5 0

- 4 0

- 3 0

- 2 0

FWM

effici

ency

[dB]

D e t u n u n g [ G H z ]

E x p e r i m e n t : D o w n - c o n v e r s i o n

T h e o r y : U p - c o n v e r s i o n D o w n - c o n v e r s i o n

(b)

1 0 0 1 0 0 04

6

8

1 0

1 2

1 4

Outpu

t Sign

al Po

wer [

dBm]

D e t u n u n g [ G H z ]

E x p e r i m e n t : f 1 f 2

T h e o r y : f 1 f 2

Figure 6.16: Calculated and measurement FWM efficiencies (a) and output signal powers (b) for differentdetunings. Visible asymmetries in the experimental efficiencies were created by asymmetries in the outputoptical power and additional imperfection of the experimentally available QD-SOA.

47

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(a)

1 2 7 5 1 3 0 0 1 3 2 5 1 3 5 0 1 3 7 502468

1 01 21 41 61 8

Mater

ial G

ain [d

B]

W a v e l e n g t h [ n m ]

5 0 G ( 1 3 2 2 . 2 n m / 1 3 2 2 . 5 n m ) 5 0 0 0 G ( 1 3 0 7 . 9 n m / 1 3 3 7 . 1 n m ) U n s a t u r a t e d Q D - S O A

(b)

1 2 7 5 1 3 0 0 1 3 2 5 1 3 5 0 1 3 7 5- 2 . 0- 1 . 5- 1 . 0- 0 . 50 . 00 . 51 . 01 . 52 . 0

Phas

e Shif

t [rad

]

W a v e l e n g t h [ n m ]

5 0 G ( 1 3 2 2 . 2 n m / 1 3 2 2 . 5 n m ) 5 0 0 0 G ( 1 3 0 7 . 9 n m / 1 3 3 7 . 1 n m ) U n s a t u r a t e d Q D - S O A

Figure 6.17: Calculated material gain (a) and additional phase modulation (b) for different detunings.Curves for unsaturated QD-SOA are presented for a reference.

(a)

0 2 4 6 8 1 00

2 0

4 0

6 0

8 0

1 0 0

Optic

al Po

wer [

mW]

T i m e [ p s ]

1 0 0 G H z 5 0 0 0 G H z

(b)

0 2 4 6 8 1 00 . 5 0

0 . 8 0

0 . 8 5

0 . 9 0

0 . 9 5Oc

cupa

tion P

roba

bility

T i m e [ p s ]

1 0 0 G H z : e l e c t r o n s h o l e s

5 0 0 0 G H z : e l e c t r o n s h o l e s

Figure 6.18: Calculated output optical powers (a) and occupation probabilities for QDs (b). Occupationprobabilities were calculated for QDs with transition frequencies near P2 CW signal.

Figure 6.15b presents a simulated spectrum of the output signal with FWM products for two inputCW with detuning of 500 GHz and input power of 5.5 dBm. ASE has not been included into thissimulation to receive a clearer output signal. Broadening of the CWs spectra were created by a numericalFourier-transformation of CWs with finite duration and it is the usual calculating error for discretemodeling.

6.3.1 Four-wave mixing between two continuous waves

First of all, we will show a FWM simulations between two CW signals. Both signals are symmetricalaround the gain peak wavelength, so they have the same material gain.

The main characteristic of the FWM process is the FWM conversion efficiency:

CE3 =P3(L)

P2(0)

CE4 =P4(L)

P1(0)

where P3(L) and P4(L) are the optical powers of FWM products on the output of the QD-SOA and P2(0)and P1(0) are input powers of the corresponding signals.

48

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QD level

BulkBarrier

E

ρ

E

ρ

Distribution of new carriers:

Fow CW signalsFow pulses

QD-FCA TPA

Figure 6.19: Schematic description of the saturations of TPA and QD-FCA processes for CW signals.CW signals create carriers in the QW in a narrower energy range, compared to broadband short pulses,this leads to a decrease of the empty energy levels in the QW for new transitions and therefore decreaserates of this processes.

Figure 6.16a shows FWM-efficiencies for CW signals for up-conversions (f4 product) and down-conversions (f3 product) and output optical powers for different detunings. Calculated efficiencies havethe same profile as the experimental ones, but smaller values. This difference based on the experimentallyavailable QD-SOA degrade over time, as all parameters in the model were fitted to conform previousexperiments. Visible asymmetry in the experiments is based on the imperfection of the experimentalmethods as well as asymmetry in the output signal powers.

For smaller detuning, decrease of the conversion efficiency is about 10 dB/decade for 10-100 GHzand 15 dB/decade for 100-1000 GHz ranges, that is significantly better, than for the bulk SOAs (near20 dB/decade). Slower decreasing of the efficiency is based on the ultrafast intradot carrier dynamic,that have time scales 0.5-1 ps.

After 1 THz we can see significant (40+ dB/decade) falling of the efficiency curves. The reasons are:- carriers are not fast enough to run with the optical power beating.- both CW signals are far away from each other and are amplified by different QDs. So there is nodynamic in the modulation of carrier density.- the gain for FWM coproducts is negligible.

As visible in Fig. 6.16b output signal powers (P1 and P2) do not significantly depends on detuning.The unsaturated gain is significantly lower for high detunings due to the inhomogeneous broadening,but in this case both signals are amplificated by different QDs, which reduce spectral hole burning. Forsmall detuning, both signals are located inside the homogeneous broadening of a single QD-group, whichproduces higher total gain saturation for closely spaced frequencies. This is visible in Fig. 6.17.

The main FWM mechanism is the material gain and refractive index grating in the semiconductormedium due to the beating between input signals. Figure 6.18a present a relation between the opticalpowers and the occupation probabilities for different detunings.

Figure 6.18b shows the carrier dynamics in a QD-group with GS transition frequency near the leftsignal. As visible, carrier density pulsations due to the beating between two CW signals are negligiblefor small detuning. For high detunings they are even smaller. The beating frequency is higher, ratherthan carriers dynamic rates, so carriers can not move as fast as optical power. The second reason - thedetuning is higher rather than the homogeneous broadening and both input signals are amplificated bydifferent QD-groups, which reduce cross-modulation.

As visible, for higher detuning we obtain significantly smaller gain pulsations, as detuning is largerthan the carrier transition rates. Less gain pulsation leads to the more constant gain over time anddramatically reduce FWM efficiency for detunings higher then 1 THz.

The both calculated FWM products (P3 and P4) are identical, because gain modulations are thesame for them and phase modulation has the same value but different signs. The modulation efficiencyis decreasing as the detuning frequency exceeds carrier transition rates as shown by many publicationsfor QD- and bulk SOAs.

49

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(a)

5 0 0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0- 5 0- 4 5- 4 0- 3 5- 3 0- 2 5- 2 0- 1 5- 1 0

FWM

effici

ency

[dB]

D e t u n u n g [ G H z ]

A l l e f f e c t s : U p - D o w n -

M a t e r i a l G a i n : G a n d Q D - F C A : U p - U p - D o w n - D o w n -

G a n d T P A : G a n d Q W - F C A : U p - U p - D o w n - D o w n -

(b)

5 0 0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0123456789

1 01 1

Outpu

t Sign

al Po

wer [

dBm]

D e t u n u n g [ G H z ]

A l l e f f e c t s : f 1 f 2

M a t e r i a l G a i n : G a n d Q D - F C A : f 1 f 1 f 2 f 2

G a n d T P A : G a n d Q W - F C A : f 1 f 1 f 2 f 2

Figure 6.20: Calculated FWM efficiencies from involving different effects for pulses sequences (a) andcorresponding output signal powers (b).

(a)

0 2 4 6 8 1 0 1 20 . 00 . 20 . 40 . 60 . 81 . 01 . 21 . 41 . 61 . 82 . 0

Optic

al Po

wer [

W]

T i m e [ p s ]

5 0 0 G H z 5 0 0 0 G H z

(a)

1 2 6 0 1 2 8 0 1 3 0 0 1 3 2 0 1 3 4 0 1 3 6 0 1 3 8 0 1 4 0 0- 6 0

- 4 0

- 2 0

0

Inten

sity [

dBm/

0.2 nm

]W a v e l e n g t h [ n m ]

5 0 0 G 5 0 0 0 G

Figure 6.21: Calculated total output optical powers for different detunings (a) and corresponding spectra(b). Fourier transformation of the calculated output signal with a 1000 GHz detuning between pulsessequences (λ1=1319.4 nm, λ2=1325.3 nm). Both signals and FWM-products are very good visible.Amplified spontaneous emission is not included into this simulation to receive more clear result.

For significant CW optical powers both QD-FCA and TPA processes are saturated, as presented onfig. 6.19, so they do not play any role and should not be included into final simulations. The values ofthis saturations are unknown and should be obtained later. For CW signals, carrier transitions QD→QWcreate significant carriers population on the energies EeGS(ES) +Ephoton and EhGS(ES)−Ephoton that leadsto the saturation of this process and reduces its efficiency. In the same way, TPA of two CW signalscreates three spectral lines in both conduction and valence bands, so they decrease the number of availablefor this process carriers and free places for carrier transitions. For using pulse sequences, signal spectraare broadened and have less peak intensity, so saturation of FCA and TPA is supposed to be negligible.

6.3.2 Four-wave mixing between two pulse sequences

A FWM between input pulses sequences is more interesting as it involves more physical effects. Figure 6.20shows calculated FWM efficiencies for simulations with different processes included. Efficiency can beseparated into 2 main parts - up to 1 THz, where we have significant carrier dynamic modulation andbeyond 1 THz where only instantaneous nonlinear processes (like TPA) are important. Output powersare shown in fig. 6.21. It is visible, that both output signals have almost the same output power. Theexception is only for including TPA, which significantly reduces the optical power for short pulses.

FWM efficiency created by material gain only is similar to the one for CW signals and has the sameprofile, except of a higher efficiency due to larger carrier dynamics. Carrier dynamics are presented infigure 6.22.

QW-FCA does not play any significant role as its influence is much smaller as compared to the othereffects. In a contrast to it, QD-FCA creates a significant asymmetry between up- and down-conversion

50

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(a)

0 2 0 4 0 6 0 8 0 1 0 0 1 2 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0

Occu

patio

n Pro

babil

ity

T i m e [ p s ]

5 0 0 G : 5 0 0 0 G : G S - e l e c t r o n s G S - e l e c t r o n s G S - h o l e s G S - h o l e s E S - e l e c t r o n s E S - e l e c t r o n s E S - h o l e s E S - h o l e s

(b)

0 2 0 4 0 6 0 8 0 1 0 0 1 2 0

2

3

4

5

6

Carri

er De

nsity

[1015

1/m2 ]

T i m e [ p s ]

5 0 0 G : 5 0 0 0 G : G S - e l e c t r o n s G S - e l e c t r o n s G S - h o l e s G S - h o l e s E S - e l e c t r o n s E S - e l e c t r o n s E S - h o l e s E S - h o l e s

Figure 6.22: Calculated carrier dynamics for GS (a) and ES (b) for a single QD-group (with transitionwavelength 1319 nm for detuning 500 GHz and 1319 nm for detuning 500 GHz and 1307 nm for detuning5000 GHz and).Carrier dynamics in QW and total carrier density in the last segment of QD-SOA.

products and this asymmetry is increasing for higher detuning between input signals. These effects canbe experimentally observed [6]. It significantly increases up-conversion and decreases down-conversionfor detunings above 1000 GHz. FCA is less important for FWM as compared to pump-probe simulations,as we have smaller carrier density pulsations over time and induced phase shift is more constant. Butdue to the fact that the gain-phase modulation “shifts” total phase response up and this makes differentmodulations for up- and down-products.

TPA and corresponding nonlinear phase modulations are extremely important in operations withpowerful pulses, as the peak optical power can be much more then 1 W. These effects are most importantfor detuning frequencies 1000 GHz and above, where they are the main mechanism, which createsnonlinear FWM products. But it is not important for detunings below 100 GHz, as carrier modulationis the most important one.

Influence of different effects

We will compare FWM efficiencies from different effects compared to only GS-gain as a reference.Important to notice that all affects nonlinearly interact with each other and the total efficiency is not asimple summation over them, but something more complex.

ES gain increases the total efficiency as we obtain additional phase modulation from ES carriers. EScarriers have a dynamic time constant similar to the GS carriers, so qualitatively both curves are almostidentical. The main difference (5 dB) is obtained for detunings near 500 GHz.

QW-FCA does not change FWM-efficiency for CW signals as we do not have carrier modulation onQW and induced phase shift is constant over time.

QD-FCA is very interesting. For small detuning the up-conversion product is higher, rather thandown conversion, but this difference is decreasing and both of them are equal for detuning of 500 GHz.For higher detuning the up-conversion product is significantly decreasing. The reason of this shift is thedelay between the peak of optical power beating and the maximum carrier depletion. Carrier densitieshave slower modulation, compared to the optical phase and they create their own “beating”. Below500 GHz QD-FCA and refractive index modulation create constructive phase modulation for up-product,after 500 GHz they create destructive modulation (they have different sign) and reduce up-conversionefficiency.

TPA and Kerr-effect are nonlinear effects and are based only on the value of the optical power. Fordetunings below 100 GHz it just increases the total efficiency by 2-3 dB, but for further detunings thiseffect became the main FWM mechanism and produces almost constant and flat profiles if the efficiencyabove 500 GHz.

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6.4 Propagation of ultrashort pulses.

One of the first experiments with a propagation of ultrashort pulses through a QD-SOA have presenteda very fast gain recovery (∼100 fs) [106], like fig. 6.23, that was not achieved by experiments presentedabove (section 6.1). In this section we will briefly discuss this difference.

Figure 6.23: Experimental pump-induced gain change for a single 150 fs pulse propagation through 475µm QD-SOA length (maximum gain 1.85 dB) [106].

The main reasons of slower gain recovery for picosecond pulses are:- pulses are shorter than the intradot carrier relaxation (500-1000 fs).- TPA is more important, as short pulse have higher peak optical power.Figure 6.24 presents calculated output optical powers and gain compressions for the single pulse

propagation through the unsaturated QD-SOA. Both pulses have the same energy (1 pJ) and differentdurations (200 fs and 1.5 ps FWHM). Both pulses are delayed to have maximum optical power in thesame time moment. It is visible that for the shorter pulse we have significantly deeper gain saturationand faster gain recovery.

Short pulses have significantly higher optical power, that leads to a higher instantaneous absorptionin the QD-SOA due to TPA.

Figure 6.25 presents intradot carrier dynamics for both pulses. Short pulses are faster than theintradot carrier relaxation time, this means that we observe smaller intradot relaxation (ES→GS) duringthe pulse. This decreases material gain and stimulated recombination, but, as we have more carriers onES, there is significantly increased relaxation after the pulse and material gain recovery.

So the utilization of ultrashort pulses allow achieving faster gain recovery, but lower amplification.Another important aspect is the spectral broadening of the pulses. As the FWHM of 200 fs pulses

is 5 THz, it is near the inhomogeneous distribution of QDs (∼7.5 THz). So the pulses are amplified bymore dots more uniformly without a significant spectral hole.

Effective amplification of shorter pulses (<200 fs) is not possible, as QDs will amplify only a smallportion of photons that have resonance frequencies equal to QDs. All other photons will propagatewithout amplification. This will lead to a significant distortion of the signal spectrum (making it morenarrow) and increases pulse duration.

Figure 6.26 present calculated carrier distributions in the inhomogeneous broadening for the lastsegment of the QD-SOA. As visible, shorter pulses produce higher spectral hole burning and are amplifiedby QDs with resonance frequency near the central signal frequency. Short pulses have a broad spectrumand are amplified by more QDs uniformly.

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- 2 0 2 4 6 80

1

2

3( a )

Outpu

t opti

cal p

ower

[W]

T i m e [ p s ]

1 5 0 0 f s 2 0 0 f s

- 2 0 2 4 6 8- 2 5

- 2 0

- 1 5

- 1 0

- 5

0( b )

Relat

ive G

ain C

ompr

essio

n [dB

]

T i m e [ p s ]

1 5 0 0 f s 2 0 0 f s

Figure 6.24: Calculated output optical powers (a) and gain compressions (b) for propagation of singlepulses (200 fs and 1500 fs FWHM) throw the unsaturated QD-SOA.

- 2 0 2 4 6 80 . 5

0 . 6

0 . 7

0 . 8

0 . 9

1 . 0( a )

Occu

patio

n Prob

abilit

y

T i m e [ p s ]

G S _ e G S _ h E S _ e E S _ h

- 2 0 2 4 6 80 . 20 . 30 . 40 . 50 . 60 . 70 . 80 . 91 . 0( b )

Occu

patio

n Prob

abilit

y

T i m e [ p s ]

G S _ e G S _ h E S _ e E S _ h

Figure 6.25: Calculated intradot occupation probabilities for propagation of 200 fs (a) and 1500 fs (b)pulses.

1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 60 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0

1 . 2

Carri

er De

nsitie

s [10

14 m

-2 ]

Q D N u m b e r

u n s a t u r a t e d : e h

2 0 0 f s : 1 5 0 0 f s : f _ G S _ e _ m i n f _ G S _ e _ m i n f _ G S _ h _ m i n f _ G S _ h _ m i n

Figure 6.26: Calculated carrier distributions in the inhomogeneous broadening for the last segment ofthe QD-SOA. Minimum (”min”) and maximum (”max”) values of occupation probabilities for electrons(”e”) and holes (”h”) for GS are presented. Unsaturated distribution is presented for a comparison.

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6.5 Improvement of linear amplification

One of the most important aspect for linear amplification is the saturation power. Increasing of itwill allow to amplify signals in more broad power range without significant distortions. Material gainsaturation occurs due to the carrier depletion in the active region and the main methods to improve itare:

- increase the carrier injection (injection current)- decrease stimulated recombination.As the injection current should be set to the maximum allowed value it is not possible to increase it

further without device damaging due to the heating. So we should decrease the stimulated recombinationfor the same optical power.

This can be achieved by decreasing the linear gain. The most simple way to create a QD-SOA withsmaller linear gain is to decrease the number of QD layers. To keep the unsaturated gain at the samevalue, the QD-SOAs should be longer.

Another possibility is to use an undoped QD-SOA. In this case we will have smaller amount of holesin GS and therefore smaller material gain.

In this section we will compare theoretical simulations for the QD-SOAs for CW and pulses. Themost important behavior for the linear amplification is in the last sections of the QD-SOA because weobtain highest saturation in it, as it was described in the CW-section. So simulations were provided forshort QD-SOAs with high input optical power to simulate the behavior under significant saturation.

Figure 6.27a presents the values of the net gain versus output optical power for reference p-dopedQD-SOA with 10 QD-layers (length 0.25 mm), p-doped QD-SOAs with 5 and 2.5 QD-layers (lengths0.61 mm and 2.49 mm) as for undoped QD-SOAs with 10 und 5 layers (lengths 0.22 mm and 0.47 mm)for a single CW signal. Injected current density is 250 mA/mm and signal wavelength corresponds tothe maximum amplification.

Important points: for smaller linear material gain the internal loses are becoming more importantand it requires to use even longer QD-SOAs with a smaller number of layers. Undoped QD-SOAs havesignificantly smaller internal losses (150 m−1) compare to p-doped (450 m−1) as the additional amountof holes in the active region creates a significant free-carrier absorption.

As visible, for an input optical power till 10 dBm these QD-SOAs have the same gain value, while for20 dBm the best result is for 5-layers undoped QD-SOA. For further increasing of the optical power wehave a significant decrease of the net gain due to two-photon absorption.

5-layer undoped QD-SOA has the best behavior, as it has a higher injected current due to the longerlength, but the length is not long enough to make nonlinear effects the most important.

Figure 6.27b presents the values of the net gain versus output optical power for pump-probe simulationsfor reference p-doped QD-SOA with 10 QD-layers (length 0.25 mm) and for undoped QD-SOAs with 5layers (lengths 0.47 mm) for RZ and NRZ 200 GHz signals. NRZ signals are more valid for linearamplification, as their peak powers are significantly smaller.

Figure 6.28 demonstrates the calculated eye-diagrams for these two QD-SOAs for different inputoptical powers. As visible, for the longer QD-SOA we obtain slightly higher average optical power andless pattern effects till 20 dBm. For higher optical power long QD-SOA shows worst behavior due tomore important nonlinear effects.

As a longer QD-SOA has almost twice higher injected current, the saturation power is ≈3 dB higherand the eye-diagrams are better for the optical power till 20 dBm.

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- 1 0 0 1 0 2 0 3 0- 5- 4- 3- 2- 10123( a )

Gain

[dB]

I n p u t P o w e r [ d B m ]

1 0 l 5 l 2 . 5 l 1 0 l u n d o p 5 l u n d o p

1 0 1 5 2 0 2 5- 1

0

1

2

3( b )

Gain

[dB]

I n p u t P o w e r [ d B m ]

1 0 l R Z 1 0 l N R Z 5 l R Z 5 l N R Z

Figure 6.27: Calculated material gain for different QD-SOAs with the same unsaturated gain for CWsignals (a) and for 200 GHz RZ and NRZ signals (b).

Figure 6.28: Calculated eye-diagrams for different QD-SOAs (p-doped with 10 QD-layers (0.25 mm) andundoped with 5 QD-layers ( 0.47 mm)) for 200 Gbit/s RZ and NRZ signals. As visible, longer QD-SOAdemonstrate better behavior for input powers till 20 dBm.

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Chapter 7

Conclusions

The presented work discusses the main properties of the QD-SOA and their possible implementations forthe telecommunication systems.

This thesis presents a novel phenomenological model to calculate the QD-SOA behavior for differentimplementations. This model is based on conventional models, but it can operate with different types ofoptical signals and provide the correct frequency dependence of the material gain and the optical phasemodulation. This is possible due to the implementation of the digital filters to simulate the gain or theabsorption. Filters allow to simplify the model, as they provide correct relations between the materialgain and the optical phase modulations (as it should be by Kramer-Kronig relations), but it does notrequire additional artificial parameters, like the alpha-factor. Values of the induced phase modulationfrom each QD group in the model depend only on the value of material gain (that is based on carrierdensity) and homogeneous broadening for this QDs group. This allows to calculate phase modulation fora single signal or the superposition of multiple signals properly, as well as to calculate material gain fordifferent spectral components of a signal properly. This makes it possible to calculate the propagation ofthe ultrashort pulses (with broader spectrum, rather than inhomogeneous broadening) as well as multiplesignals and automatically receive all gain and phase based effects. Other conventional models may providethe material gain that is equal for all spectral components, which limits their utilization to signals withlow bitrate and narrow spectrum. Phase modulation in these models is created by additional coefficients,which can be strictly considered only for the central frequency of a single signal.

The first part of this thesis presents the theoretical model and the second part demonstrates thecalculated results and their comparison with the experimentally provided data. As the model alreadyincludes all required effects it does not require to change calculation parameters for different types ofsimulations and we obtain all effects at the same time for any simulation. The theoretical and experimentalresults for different types of measurements are compared quantitatively and are in good agreement. Thisverifies the presented model and permits its utilization in the future QD-SOA development.

The presented model has the modular structure, as presented in section 3.5. Each effect is calculatedseparately and depends only on the carrier densities and optical power for every length segment and timemoment. This allows to determinate the influence of each effect for every simulation as well as add anynew effect in the model easily.

This model can be used for any type of QD-SOA (as well as for QD CW laser with some improvementsand gain sections of the mode-locked lasers) with different transition wavelengths, as these devices havethe same active medium and the same physical effects. Only correction of the coefficients will be requiredas different QD-SOAs have different QDs and therefore slightly different behaviors.

The QD-SOAs behavior is based on the physical properties of QDs and modeled at most fit case formodeling of the whole telecommunication system.

The values of all coefficients and parameters are based on the experiments for a single QD-SOA, thatallows to analyze the QD-SOA behavior in different regimes. Some effects can be neglected in one typeof experiment and simulation, but are very important for other ones. Only such complex analysis can

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give the complete picture of physical processes in the QD-SOA device. And allows to use this model insubsequent utilizations.

The main benefits of the QD-SOAs compared to bulk and QW structures are:

• High saturation power, which is required for linear amplification of different signals, as presentedin the section 7.5. This can be achieved due to the discrete energy levels of the QDs, where thematerial gain depends only on the carrier density on the first energy level and all upper statesand QW are working as a carrier reservoir. The other reason is the smaller linear gain, as QDactive region has smaller number of the recombination centers per unite volume compare to higherdimensional structures. To receive the same value of the unsaturated gain it is required to create alonger SOA, that increase the total injected current for the same injected current density.

• Ultrafast gain recovery, as presented in sections 6.1 and 6.4, due to the inradot carrier dynamic.

• Chirp-free linear amplification for small optical power, as presented in sections 5.2 and 6.4. Asthe induced phase modulation due to the material gain has zero value for the central wavelengthof the inhomogeneous distribution it is possible to amplify phase modulated signal with minimaldistortions. In addition, by selecting the optical wavelengths it is possible to obtain positive ornegative values of the chirp. For high optical power the values of the phase modulation are almostequal for broad range of the signal wavelengths.

• For high optical power the total value of the material gain has minor frequency dependence comparedto the low optical power, as presented in section 6.2. This allows to achieve the similar values ofthe material gain for the optical signals with different central wavelengths.

• High four-wave mixing efficiency, as presented in section 6.3. The four-wave mixing is based on fastintradot relaxation times and its values are higher, rather then for bulk- or QW-structures.

The further improvement of the presented model depends on the required applications. The mainpoints, which should be investigated in more detaile by the experiments are:

• Shape of the inhomogeneous broadening and its dependence on the carrier densities and activeregion temperature.

• Shape of the homogeneous broadening and its dependence on the carrier dynamics.

• More detailed carrier dynamic that is based on QW carrier distributions as well as on active regiontemperatures.

• Add polarization dependence of the signal if it will be required. This effect is based on the activeregion size and does not have directly interaction with QDs.

The presented model may be used for QD-SOAs as well as for modeling of QD lasers and gainsections of the mode-locked lasers. Due to the modular structure of the model it will be possible to addnew physical effects if it will be required.

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7.1 Acknowledgements

First of all, I would like to thank Prof. Klaus Petermann for the opportunity to work on my Ph.D. thesisin his dream-team and scientific guidance.

I would like to thank the Deutsche Forschungsgemeinschaft (German Research Foundation) for thepoject funding.

I would also like to acknowledgement Dr. Carsten Schmidt-Langhorst, Dr. Colja Schubert and Andrey Galperinfrom the Fraunhofer Institute for Telecommunications (Heinrich Hertz Institute) and Dr. Christian Meuerfor the providing of the experimental results to verify my theoretical model.

In addition, I am thankful to my colleags Dr. Christian-Alexander Bunge, Dr. Patrick Runge andDr. Robert Elschner for the productive help on the early stage of my work and Abdul Rahim for helpingwith my english.

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Appendix A

Calculation of intraband transitionrates

Intraband transition rates are calculated by the following equations [20]:

SeQW−ES =− 80 · (neQW · 10−24m2)7 + 2904 · (neQW · 10−24m2)6 − 43990 · (neQW · 10−24m2)5

+ 366613 · (neQW · 10−24m2)4 − 1799800 · (neQW · 10−24m2)3 + 6920515 · (neQW · 10−24m2)2

+ 796801 · (neQW · 10−24m2)

SeQW−GS =− 55 · (neQW · 10−24m2)7 + 1996 · (neQW · 10−24m2)6 − 30476 · (neQW · 10−24m2)5

+ 256293 · (neQW · 10−24m2)4 − 1275342 · (neQW · 10−24m2)3 + 4751747 · (neQW · 10−24m2)2

+ 534424 · (neQW · 10−24m2)

SeES−GS =9261 · (neQW · 10−24m2)7 − 341370 · (neQW · 10−24m2)6 + 5210563 · (neQW · 10−24m2)5

− 42779164 · (neQW · 10−24m2)4 + 205945315 · (neQW · 10−24m2)3 − 598788530 · (neQW · 10−24m2).2

+ 1.129186559 · (neQW · 10−24m2)

ShQW−ES =− 32 · (nhQW · 10−24m2)71744 · (nhQW · 10−24m2)6 − 39904 · (nhQW · 10−24m2)5

+ 489580 · (nhQW · 10−24m2)4 − 3508244 · (nhQW · 10−24m2)3 + 18830750 · (nhQW · 10−24m2)2

+ 4297407 · (nhQW · 10−24m2)

ShQW−GS =− 28 · (nhQW · 10−24m2)7 + 1536 · (nhQW · 10−24m2)6 − 35148 · (nhQW · 10−24m2)5

+ 432184 · (nhQW · 10−24m2)4 − 3106695 · (nhQW · 10−24m2)3 + 16489215 · (nhQW · 10−24m2)2

+ 0.003591389 · (nhQW · 10−24m2)

ShES−GS = + 3634 · (nhQW · 10−24m2)7 − 196943 · (nhQW · 10−24m2)6 + 4380417 · (nhQW · 10−24m2)5

− 51609924 · (nhQW · 10−24m2)4 + 346820886 · (nhQW · 10−24m2)3 − 1328189900 · (nhQW · 10−24m2)2

+ 2.608732836 · (nhQW · 10−24m2)

(A.1)

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Appendix B

Acronyms

Table B.1: List of Acronyms.0D zero dimentional1D one dimentional2D two dimentional3D three dimentionalASE amplified spontaneous emissionCW continuous waveDOS density of statesDWELL dots in a wellES excited stateFCA free-carriers absorptionFWHM full width half maximumFWM four-wave mixingGS ground stateIIR infinite impulse responseNRZ non return to zeroPP pump-probeQD quantum dotQW quantum wellRZ return to zeroSOA semiconductor optical amplifierTD time-domainTPA two-photon absorptionSE spontaneous emissionWDM wavelength division multiplexingXGM cross-gain modulationXPM cross-phase modulation

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Appendix C

Coefficients

Table C.1: List of parameters and coefficients used in the simulationsParameter Value QuantityL 2 mm SOA lengthW 4 µm Active region widthH 0.45 µm Active region heightNlay 10 Number of QDs layersM 16 Number of QDs groupsnref 3.5 Group delay indexD2D 2·1015 m−2 2D QD densityNdop 4·1015 m−2 2D p-doping densitydt 20 fs Calculation time stepdz 1.71 µm Calculation length stepf0 237 THz Central frequency of the modelEES,0 1016 meV ES central transition energyEGS,0 946 meV GS central transition energy

Ee(h)GS,0 273 (-73) meV GS central energy

Ee(h)ES,0 930.6 (-85.4) meV ES central energy

Ee(h)QW 1063.1 (-186.9) meV Energy of the first QW level

me(h) 0.025(0.074) Effective electron(hole) massεGS(ES) 2(4) Density of states for GS(ES)γ0 10 meV FWHM of homogeneous broadeningηGS(ES),0 35 meV FWHM of inhomogeneous broadeningkRSGS(ES) 4.58 ·10−5(1.63 ·10−5) A−1 Red-shift coefficient

kRS∆E 0.45 SE filter coefficient∆EGS 4 meV Energy separation between QD groupsag 1125 m−1 Linear gainΓ 0.3 Gain confinementkSE 22.9 SE filter coefficientβSE 5.3·10−3 SE coupling coefficientBQW 850·10−9 s·m2 QW spontaneous recombination coefficientτES = τGS 2 ns QD spontaneous recombination timeγTPA 580 m−1W−1 TPA coefficientb2 430 m−1W−1 Self-phase modulation termαint.los. 450 m−1 Internal losses

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Appendix D

List of main symbols

Symbol DescriptionA Filter coefficientag Linear gain parameterb2 Nonlinear phase modulation termBQW QW spontaneous recombination coefficientB Filter coefficientCDrude Drude coefficientD(m) the fraction of QDs in subgroup mD2D 2D QD densitydt Calculation time stepdz Calculation length stepEES,0 ES central transition energyEGS,0 GS central transition energy

Ee(h)GS GS central energy

Ee(h)ES ES central energy

Ee(h)QW Energy at QW band edge

E(t) Normalised complex envelope of the electric field∆EGS Energy separation between QD groupsEGS Energy separation between QD groupsf0 Central frequency of the model

fe(h)GS(ES)(m) occupation probability for m-th QD group

fS signal frequencyG Linear material gaing Linear material gain coeffitientH Active region heightH(Z) Impulse response in the Z domainhm(t) Impulse response of m-th filter

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Symbol DescriptionH(f) Frequency response of a filterk wave vectorkRSGS(ES) Red-shift coefficient

kSE SE filter coefficientkRS∆E SE filter coefficientL SOA lengthLx Length of the semiconductor crystal in x dyrectionM Number of QDs groupsme(h) Effective electron(hole) mass

ne(h)QW 2D carrier density in QW

Ndop 2D p-doping densityN3De(h) 3D carrier density

ne(h)QW 2D electrons (holes) density for QW

Nlay Number of QDs layersnref Group delay indexP Optical powerRstim(spont) Value of stimulated (spontaneous) recombination

Se(h)QW−ES QW-ES capturing rate for electrons (holes)

Se(h)QW−GS QW-GS capturing rate for electrons (holes)

Se(h)ES−GS ES-GS relaxation rate for electrons (holes)T Active region temperatureT2 Dephasing timeW Active region widthZ−1 Delay element in the Z domainZG(SE) Delay element for gain (spontaneous emission)αint.los. Internal lossesεGS(ES) Density of states for GS(ES)γ0 FWHM of homogeneous broadeningΓ Optical confinement factorηGS(ES),0 FWHM of inhomogeneous broadeningηbg Background refractive index∆ηFCA Refractive index modulation due to FCAΓ Gain confinementβSE SE coupling coefficientηinj Injected current efficiencyτES = τGS QD spontaneous recombination timeγTPA TPA coefficientρ Density of statesϕρ Optical phaseω Angular frequency

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Bibliography

[1] D. Bimberg, M. Grundmann, N. N. Ledentsov. Quantrm Dot Heterostructures. John Willey & Sons,1999.

[2] L. Jacak, P. Hamrylak, A. Wojs. Quantrm Dots. Springer, 1999.

[3] T. W. Berg. Quantum Dot Semiconductor Optical Amplifiers, Ph.D. thesis. Research Center COM,Technical University of Denmark, 2004.

[4] A. J. Zilkie. High-speed properties of 1.55-µm-wavelength quantum dot semiconductor amplifiersand comparison with higher-dimentional structures, Ph. D. Thesis. University of Toronto, Toronto,2008.

[5] T. W. Berg, J. Mørk. Saturation and noise properties of quantum-dot optical amplifiers. IEEE J.Quantum Electron., 40, no. 11:1527–1539, 2004.

[6] D. Nielsen, S. L. Chuang. Four-wave mixing and wavelength conversion in quantum dots. Phys.Rev. B, 81:035305 (11 pp.), 2010.

[7] O. Qasaimeh. Novel Closed-Form Model for Multiple-State Quantum-Dot Semiconductor OpticalAmplifiers. IEEE J. Quantum Electron., 44, no. 7:652–657, 2008.

[8] A. Markus. Impact of intraband relaxation on the performance of a quantum-dot laser. IEEE J.Select. Topics Quantum Electron., 9, no. 5:1308–1314, 2003.

[9] T. W. Berg and J. Mørk. Quantum dot amplifiers with high output power and low noise. Appl.Phys. Lett., 82, no. 18:3083–3095, 2003.

[10] A. J. Zilkie, J. Meier, M. Mojahedi, P. J. Poole,P. Barrios, D. Poitras, T. J. Rotter, Chi Yang,A. Stintz, K. J. Malloy, P. W. E. Smith, J. S. Aitchison. Carrier Dynamics of Quantum-Dot,Quantum-Dash, and Quantum-Well Semiconductor Optical Amplifiers Operating at 1.55 µm. IEEEJ. Quantum Electron, 43, no. 11:982–991, 2007.

[11] T. W. Berg, S. Bischoff, I. Magnusdottir, J. Mørk. Ultrafast gain recovery and modulationlimitations in self-assembled quantum-dot devices. IEEE Photon. Technol. Lett., 13, no. 6:541–543, 2001.

[12] M. van der Poel, E. Gehrig, O. Hess, D. Birkedal, J. M. Hvam. Ultrafast Gain Dynamics inQuantum-Dot Amplifiers: Theoretical Analysis and Experimental Investigations. IEEE J. QuantumElectron., 41, no. 9:1115–1123, 2005.

[13] A. V. Uskov, T. W. Berg, J. Mørk. Theory of pulse-train amplification without patterning effectsin quantum-dot semiconductor optical amplifiers. IEEE J. Quantum Electron., 40, no. 3:306–320,2004.

[14] A. Fiore, A. Markus. Differential Gain and Gain Compression in Quantum-Dot Lasers. IEEE J.Quantum Electron., 43, no. 4:287–294, 2007.

69

Page 70: Modeling of Quantum Dot Semiconductor Optical … of Quantum Dot Semiconductor Optical Ampli ers for Telecommunication Networks vorgelegt von Diplom-Ingenieur Dmitriy Puris von der

[15] A. A. Dikshit, J. M. Pikal. Carrier distribution, gain, and lasing in 1.3-µm InAs-InGaAs quantum-dot lasers. IEEE J. Quantum Electron., 40, no. 2:105–112, 2004.

[16] C. Z. Tong, S. F. Yoon, C. Y. Ngo, C. Y. Liu, W. K. Loke. Rate Equations for 1.3-µm Dots-Under-a-Well and Dots-in-a-Well Self-Assembled InAsGaAs Quantum-Dot Lasers. IEEE J. QuantumElectron., 42, no. 11:1175–1183, 2006.

[17] X. Jin-Long Xiao, Y.-Z. Huang. Numerical Analysis of Gain Saturation, Noise Figure, and CarrierDistribution for Quantum-Dot Semiconductor-Optical Amplifiers. IEEE J. Quantum Electron., 44,no. 5:448–455, 2008.

[18] J. Gomis-Bresco, S. Dommers, V. V. Temnov, U. Woggon, J. Martinez-Pastor, M. Laemmlin,D. Bimberg. InGaAs Quantum Dots Coupled to a Reservoir of Nonequilibrium Free Carriers.IEEE J. Quantum Electron., 45, no. 9:1121–1128, 2009.

[19] S. Dommers, V. V. Temnov, U. Woggon, J. Gomis, J. Martinez-Pastor, M. Laemmlin, andD. Bimberg. Complete ground state gain recovery after ultrashort double pulses in quantum dotbased semiconductor optical amplifier. Appl. Phys. Lett., 90:033508 (3 pp.), 2007.

[20] N. Majer, K. Ludge, and E. Scholl. Cascading enables ultrafast gain recovery dynamics of quantumdot semiconductor optical amplifiers. Phys. Rev. B, 82:235301 (6 pp.), 2010.

[21] J. Kim, S. L. Chuang. Theoretical and experimental study of optical gain, refractive index change,and linewidth enhancement factor of p-doped quantum-dot lasers. IEEE J. Quantum Electron., 42,no. 9:942–952, 2006.

[22] J. Kim, M. Laemmlin, C. Meuer, D. Bimberg, G. Eisenstein. Static Gain Saturation Model ofQuantum-Dot Semiconductor Optical Amplifiers. IEEE J. Quantum Electron., 44, no. 7:648–666,2008.

[23] J. Kim, M. Laemmlin, C. Meuer, D. Bimberg, G. Eisenstein. Theoretical and ExperimentalStudy of High-Speed Small-Signal Cross-Gain Modulation of Quantum-Dot Semiconductor OpticalAmplifiers. IEEE J. Quantum Electron., 45, no. 3:240–248, 2009.

[24] C. Meuer, J. Kim, M. Laemmlin, S. Liebich, G. Eisenstein, R. Bonk, T. Vallaitis, J. Leuthold,A. Kovsh, I. Krestnikov, D. Bimberg. High-Speed Small-Signal Cross-Gain Modulation in Quantum-Dot Semiconductor Optical Amplifiers at 1.3 µm. IEEE J. Select. Topics Quantum Electron., 15,no. 3:749–756, 2009.

[25] D. Bimberg, N. Kirstaedter, N. N. Ledentsov, Zh. I. Alferov, P. S. Kop’ev, V. M. Ustinov. InGaAs-GaAs quantum-dot lasers. IEEE J. Select. Topics Quantum Electron., 3, no. 2:196–204, 1997.

[26] P. Blood. Gain and Recombination in Quantum Dot Lasers. IEEE J. Select. Topics QuantumElectron., 15, no. 3:808–817, 2009.

[27] J. Marquez, L. Geelhaar, and K. Jacobi. Atomically resolved structure of InAs quantum dots. Appl.Phys. Lett., 78:2309 (3 pp.), 2001.

[28] L. He and A. Zunger. Multiple charging of InAs/GaAs quantum dots by electrons or holes: Additionenergies and ground-state configurations. Phys. Rev. B, 73:115324 (14 pp.), 2006.

[29] H. Jiang, J. Singh. Self-assembled semiconductor structures: electronic and optoelectronicproperties. IEEE J. Quantum Electron., 34, no. 7:1188–1196, 1998.

[30] P. Blood. On the dimensionality of optical absorption, gain, and recombination in quantum-confinedstructures. IEEE J. Quantum Electron., 36, no. 3:354–362, 2000.

70

Page 71: Modeling of Quantum Dot Semiconductor Optical … of Quantum Dot Semiconductor Optical Ampli ers for Telecommunication Networks vorgelegt von Diplom-Ingenieur Dmitriy Puris von der

[31] O. Stier, M. Grundmann, and D. Bimberg. Electronic and optical properties of strained quantumdots modeled by 8-band k·p theory. Phys. Rev. B, 59:5688–5701, 1999.

[32] A. Lenz. Atomic structure of capped In(Ga)As and GaAs quantum dots for optoelectronic devices,Ph.D. Thesis. Technische Universitat Berlin, 2008.

[33] W. W. Chow, S. W. Koch. Semiconductor-Laser Fundamentals. Springer, 1999.

[34] M. Gong, K. Duan, C.-F. Li, R. Magri, G. A. Narvaez, and L. He. Electronic structure of self-assembled InAs/InP quantum dots: Comparison with self-assembled InAs/GaAs quantum dots.Phys. Rev. B, 77:045326 (10 pp.), 2008.

[35] W. Sheng and P. Hawrylak. Atomic structures of electronic and optical properties of InAs/InPself-assembled quantum dots on patterned substrates. Phys. Rev. B, 72:035326 (8 pp.), 2005.

[36] C. Cornet, A. Schliwa, J. Even, F. Dore, C. Celebi, A. Letoublon, E. Mace, C. Paranthoen, A. Simon,P. M. Koenraad, N. Bertru, D. Bimberg, and S. Loualiche . Electronic and optical properties ofInAs/InP quantum dots on InP(100) and InP(311)B substrates: Theory and experiment. Phys.Rev. B, 74:035312 (9 pp.), 2006.

[37] A. Galperin. Experimentelle Untersuchung der nichtlinearen Verstarkungsdynamik vonQuantenpunkt-Halbleiterlaserverstarkern mittels optiacher Pikosekundenpulse, Diplom Thesis.Technische Universitat Berlin, 2009.

[38] E. Malic, M. J. P. Bormann, P. Hovel, M. Kuntz, D. Bimberg, A. Knorr, E. Scholl. CoulombDamped Relaxation Oscillations in Semiconductor Quantum Dot Lasers. IEEE J. Select. TopicsQuantum Electron., 13, no. 5:1242–1248, 2007.

[39] B. Ohnesorge, M. Albrecht, J. Oshinowo, and A. Forchel. Rapid carrier relaxation in self-assembledInxGa1−xAs/GaAs quantum dots. Phys. Rev. B, 54:11532 (7 pp), 1996.

[40] A. V. Uskov, J. McInerney, F. Adler, H. Schweizer, and M. H. Pilkuhn. Auger carrier capturekinetics in self-assembled quantum dot structures. Appl. Phys. Lett., 72:58 (3 pp), 1998.

[41] S. Raymond, K. Hinzer, S. Fafard, J. L. Merz. Experimental determination of Auger capturecoefficients in self-assembled quantum dots. Phys. Rev. B, 61:6331–16334, 2000.

[42] J. Gomis-Bresco, S. Dommers, V. V. Temnov, U. Woggon, E. Malic, M. Richter, E. Scholl, andA. Knorr. Impact of Coulomb Scattering on the Ultrafast Gain Recovery in InGaAs Quantum Dots.Appl. Phys. Lett., 101:256803 (3 pp.), 2007.

[43] P. Borri, W. Langbein, S. Schneider, U. Woggon, R. L. Sellin, D. Ouyang, D. Bimberg. Excitonrelaxation and dephasing in quantum-dot amplifiers from room to cryogenic temperature. IEEE J.Select. Topics Quantum Electron., 8, no. 5:984–991, 2002.

[44] C. Lingk, W. Helfer, G. von Plessen, J. Feldmann, K. Stock, M. W. Feise, D. S. Citrin, H. Lipsanen,M. Sopanen, R. Virkkala, J. Tulkki, J. Ahopelto. Carrier capture processes in strain-inducedInxGa1−xAs/GaAs quantum dot structures. Phys. Rev. B, 62:13588–13594, 2000.

[45] S. Mokkapati, M. Buda, H. H. Tan, and C. Jagadish. Effect of Auger recombination on theperformance of p-doped quantum dot lasers. Appl. Phys. Lett., 88:161121 (3 pp.), 2006.

[46] T. R. Nielsen, P. Gartner, and F. Jahnke. Many-body theory of carrier capture and relaxation insemiconductor quantum-dot lasers. Appl. Phys. Lett., 69:235314 (13 pp.), 2004.

[47] S.-W. Chang, S.-L. Chuang, and N. Holonyak, Jr. Phonon- and Auger-assisted tunneling from aquantum well to a quantum dot. Appl. Phys. Lett., 70:125312 (12 pp.), 2004.

71

Page 72: Modeling of Quantum Dot Semiconductor Optical … of Quantum Dot Semiconductor Optical Ampli ers for Telecommunication Networks vorgelegt von Diplom-Ingenieur Dmitriy Puris von der

[48] R. Heitz, M. Veit, N. N. Ledentsov, A. Hoffmann, D. Bimberg, V. M. Ustinov, P. S. Kopev, andZh. I. Alferov. Energy relaxation by multiphonon processes in InAs/GaAs quantum dots. Phys.Rev. B, 56:10435 (11 pp), 1997.

[49] J. Urayama, T. B. Norris, J. Singh and P. Bhattacharya. Observation of Phonon Bottleneck inQuantum Dot Electronic Relaxation. Phys. Rev. Lett., 86:4930 (4 pp), 2001.

[50] D. G. Deppe, H. Huang, O. B. Shchekin. Modulation characteristics of quantum-dot lasers: theinfluence of p-type doping and the electronic density of states on obtaining high speed. IEEE J.Quantum Electron., 38, no. 12:1587–1593, 2002.

[51] G. Toptchiyski, S. Kindt, K. Petermann, E. Hilliger, S. Diez, H. G. Weber. Time-domain modelingof semiconductor optical amplifiers for OTDM applications. IEEE J. Lightwave Technol., 17, no.12:2577–2583, 1999.

[52] P. Runge. Nonlinear Effects in Ultralong Semiconductor Optical Amplifiers for OpticalCommunications: Physics and Applications, Ph.D. Thesis. Technische Universitat Berlin, 2010.

[53] R. E. Bogner and A. G. Constantinides. Introduction to Digital Filtering. John Wiley and Sons,1975.

[54] I. C. Sandall, P. M. Smowton, C. L. Walker, T. Badcock, D. J. Mowbray, H. Y. Liu, andM. Hopkinson. The effect of p doping in InAs quantum dot lasers. Appl. Phys. Lett., 88:111113(3 pp), 2006.

[55] H. Ju, A. V. Uskov, R. Notzel, Z. Li, J. Molina Vazquez, D. Lenstra, G. D. Khoe, H. J. S. Dorren.Effect of two photon absorption on carrier dynamics in quantum-dot optical amplifiers. Appl. Phys.B., 82, no. 4:615–620, 2006.

[56] P. Aivaliotis, E. A. Zibik, L. R. Wilson, J. W. Cockburn, M. Hopkinson, and N. Q. Vinh. Twophoton absorption in quantum dot-in-a-well infrared photodetectors. Appl. Phys. Lett., 92:023501(3 pp.), 2008.

[57] J. Kim. Effect of Free-carrier Absorption on the Carrier Dynamics of Quantum-dot SemiconductorOptical Amplifiers. J. Korean Phys. Soc., 55 no. 2:512–516, 2009.

[58] A. V. Uskov, E. P. OReilly, D. McPeake, N. N. Ledentsov, D. Bimberg, and G. Huyet. Carrier-induced refractive index in quantum dot structures due to transitions from discrete quantum dotlevels to continuum states. Appl. Phys. Lett., 84:272 (3 pp.), 2004.

[59] C. H. Henry, R. A. Logan, and K. A. Bertness. Spectral dependence of the change in refractiveindex due to carrier injection in GaAs lasers. J. App. Phys., 52:4457 (5pp), 1981.

[60] C. H. Henry. Theory of the linewidth of semiconductor lasers. IEEE J. Quantum Electron., 18, no.2:259–264, 1982.

[61] G. P. Agrawal. Intensity dependence of the linewidth enhancement factor and its implications forsemiconductor lasers. IEEE Photon. Technol. Lett., 1, no. 8:212–214, 1989.

[62] G. P. Agrawal, N. A. Olsson. Self-phase modulation and spectral broadening of optical pulses insemiconductor laser amplifiers. IEEE J. Quantum Electron., 25 no. 11:2297–2306, 1989.

[63] J. Wang, A. Maitra, C. G. Poulton, W. Freude, J. Leuthold. Temporal Dynamics of the AlphaFactor in Semiconductor Optical Amplifiers. IEEE J. Lightwave Technol, 25, no. 3:891–900, 2007.

[64] P. Borri, W. Langbein, U. Woggon, V. Stavarache, D. Reuter, and A. D. Wieck. Exciton dephasingvia phonon interactions in InAs quantum dots: Dependence on quantum confinement. Phys. Rev.B, 71:115328 (8 pp), 2005.

72

Page 73: Modeling of Quantum Dot Semiconductor Optical … of Quantum Dot Semiconductor Optical Ampli ers for Telecommunication Networks vorgelegt von Diplom-Ingenieur Dmitriy Puris von der

[65] P. Borri, W. Langbein, J. Mørk, J. M. Hvam, F. Heinrichsdorff, M.-H. Mao, and D. Bimberg.Dephasing in InAs/GaAs quantum dots. Phys. Rev. B, 60:7784 (4 pp.), 2008.

[66] H. Kamada and T. Kutsuwa. Broadening of single quantum dot exciton luminescence spectra dueto interaction with randomly fluctuating environmental charges. Phys. Rev. B, 78:155324 (16 pp.),2008.

[67] W. Langbein, P. Borri, U. Woggon, V. Stavarache, D.+Reuter, and A. D. Wieck. Radiativelylimited dephasing in InAs quantum dots. Phys. Rev. B, 70:033301 (4 pp.), 2004.

[68] B. Krummheuer, V. M. Axt, and T. Kuhn. Theory of pure dephasing and the resulting absorptionline shape in semiconductor quantum dots. Phys. Rev. B, 65:195313 (12 pp.), 2002.

[69] P. Borri, W. Langbein, S. Schneider, U. Woggon, R. L. Sellin, D. Ouyang, and D. Bimberg. UltralongDephasing Time in InGaAs Quantum Dots. Phys. Rev. Lett., 87:157401 (4 pp.), 2001.

[70] E. A. Muljarov, and R. Zimmermann. Dephasing in Quantum Dots: Quadratic Coupling to AcousticPhonons. Phys. Rev. Lett., 93:237401 (4 pp.), 2004.

[71] C. Kammerer, C. Voisin, G. Cassabois, C. Delalande, Ph. Roussignol, F. Klopf, J. P. Reithmaier,A. Forchel, and J. M. Gerard. Line narrowing in single semiconductor quantum dots: Toward thecontrol of environment effects. Phys. Rev. B, 66:041306 (4 pp.), 2002.

[72] A. V. Uskov, K. Nishi, and R. Lang. Collisional broadening and shift of spectral lines in quantumdot lasers. Appl. Phys. Lett., 74:3081 (3 pp.), 1999.

[73] D. Birkedal, K. Leosson, and J. M. Hvam. Long Lived Coherence in Self-Assembled Quantum Dots.Appl. Phys. Lett., 87:227401 (4 pp.), 2001.

[74] A. Markus, M. Rossetti, V. Calligari, J. X. Chen, and A. Fiore. Role of thermal hopping andhomogeneous broadening on the spectral characteristics of quantum dot lasers. J. Appl. Phys.,98:104506 (8 pp.), 2005.

[75] M. Sugawara, K. Mukai, Y. Nakata, H. Ishikawa, A. Sakamoto. Effect of homogeneous broadeningof optical gain on lasing spectra in self-assembled InxGa1−xAs/GaAs quantum dot lasers. Phys.Rev. B, 61:7595 (9 pp.), 2000.

[76] M. Bayer and A. Forchel. Temperature dependence of the exciton homogeneous linewidth inIn0.60Ga0.40As/GaAs self-assembled quantum dots. Phys. Rev. B, 65:041308 (4 pp.), 2002.

[77] M. Lorke, T. R. Nielsen, J. Seebeck, P. Gartner, and F. Jahnke. Influence of carrier-carrier andcarrier-phonon correlations on optical absorption and gain in quantum-dot systems. Phys. Rev. B,73:085324 (10 pp.), 2006.

[78] G.. P. Agrawal, N. K. Dutta. Long wavelength semiconductor lasers. Van Nostrand Reinhold, 1986.

[79] T. Miyazawa, T. Nakaoka, T. Usuki, Y. Arakawa, K. Takemoto, S. Hirose, S. Okumura, M. Takatsu,and N. Yokoyama. Exciton dynamics in current-injected single quantum dot at 1.55 µm. Appl.Phys. Lett., 92:161104 (3 pp.), 2008.

[80] P. Miska, J. Even, X. Marie, and O. Dehaese. Electronic structure and carrier dynamics in InAs/InPdouble-cap quantum dots. Appl. Phys. Lett., 94:061916 (3 pp.), 2009.

[81] C. Cornet, C. Labbe, H. Folliot, P. Caroff, C. Levallois, O. Dehaese, J. Even, A. Le Corre, andS. Loualiche. Time-resolved pump probe of 1.55 µm InAs/InP quantum dots under high resonantexcitation. Appl. Phys. Lett., 88:171502 (3 pp.), 2006.

73

Page 74: Modeling of Quantum Dot Semiconductor Optical … of Quantum Dot Semiconductor Optical Ampli ers for Telecommunication Networks vorgelegt von Diplom-Ingenieur Dmitriy Puris von der

[82] S. Azouigui, B. Dagens, F. Lelarge, J.-G. Provost, D. Make, O. Le Gouezigou, A. Accard,A. Martinez, K. Merghem, F. Grillot, O. Dehaese, R. Piron, S. Loualiche, Qin Zou, and A. Ramdane.Optical Feedback Tolerance of Quantum-Dot- and Quantum-Dash-Based Semiconductor LasersOperating at 1.55 µm. J. Select. Topics Quantum Electron., 15, no. 3:754–773, 2009.

[83] P. Miska, J. Even, O. Dehaese, and X. Marie. Carrier relaxation dynamics in InAs/InP quantumdots. Appl. Phys. Lett., 92:191103 (3 pp.), 2008.

[84] R. Hostein, A. Michon, G. Beaudoin, N. Gogneau, G. Patriache, J.-Y. Marzin, I. Robert-Philip,I. Sagnes, and A. Beveratos. Time-resolved characterization of InAsP/InP quantum dots emittingin the C-band telecommunication window. Appl. Phys. Lett., 93:073106 (3 pp.), 2008.

[85] I. O’Driscoll, T. Piwonski, J. Houlihan, G. Huyet, R. J. Manning, and B. Corbett. Phase dynamicsof InAs/GaAs quantum dot semiconductor optical amplifiers. Appl. Phys. Lett., 91:263506 (3 pp.),2007.

[86] P. P. Baveja, D. N. Maywar, A. M. Kaplan, G. P. Agrawal. Self-Phase Modulation in SemiconductorOptical Amplifiers: Impact of Amplified Spontaneous Emission. IEEE J. Quantum Electron., 46no. 9:1396–1403, 2010.

[87] M. Gioannini, I. Montrosset. Numerical Analysis of the Frequency Chirp in Quantum-DotSemiconductor Lasers. IEEE J. Quantum Electron., 43 no. 10:941–949, 2007.

[88] J. Kim, C. Meuer, D. Bimberg, and G. Eisenstein. Role of carrier reservoirs on the slow phaserecovery of quantum dot semiconductor optical amplifiers. Appl. Phys. Lett., 94:041112 (3pp),2009.

[89] T. Vallaitis, C. Koos, R. Bonk, W. Freude, M. Laemmlin, C. Meuer, D. Bimberg, and J. Leuthold.Slow and fast dynamics of gain and phase in a quantum dot semiconductor optical amplifier. Opt.Expr., 16:170–178, 2008.

[90] S. Gerhard, C. Schilling, F. Gerschutz, M. Fischer, J. Koeth, I. Krestnikov, A. Kovsh, M. Kamp,S. Hofling, A. Forchel. Frequency-Dependent Linewidth Enhancement Factor of Quantum-DotLasers. IEEE Photon. Technol. Lett., 20, no. 20:1736–1738, 2008.

[91] F. Grillot, B. Dagens, J.-G. Provost, Hui Su, L. F. Lester. Gain Compression and Above-ThresholdLinewidth Enhancement Factor in 1.3-µm InAsGaAs Quantum-Dot Lasers. IEEE J. QuantumElectron., 44, no. 10:946–951, 2008.

[92] S. Schneider, S. P. Borri, W. Langbein, U. Woggon, R. L. Sellin, D. Ouyang, D. Bimberg. Linewidthenhancement factor in InGaAs quantum-dot amplifiers. IEEE J. Quantum Electron., 40, no.10:1423–1429, 2004.

[93] J. M. Vazquez, H. H. Nilsson, J.-Z. Zhang, I. Galbraith. Linewidth Enhancement Factor ofQuantum-Dot Optical Amplifiers. IEEE J. Quantum Electron., 45, no. 10:986–993, 2006.

[94] V. Cesari, V. P. Borri, M. Rossetti, A. Fiore, W. Langbein. Refractive Index Dynamics andLinewidth Enhancement Factor in p-Doped InAsGaAs Quantum-Dot Amplifiers. IEEE J. QuantumElectron., 45, no. 6:579–585, 2009.

[95] A. J. Zilkie, J. Meier, M. Mojahedi, A. S. Helmy, P. Poole, P. Barrios, D. Poitras, T. J. Rotter,Chi Yang, A. Stintz, K. J. Malloy, P. W. E. Smith, S. J. Aitchison. Time-Resolved LinewidthEnhancement Factors in Quantum Dot and Higher-Dimensional Semiconductor AmplifiersOperating at 1.55 µm. IEEE J. Lightwave Technol., 26, no. 11:1498–1509, 2008.

[96] M. Y. Hong, Y. H. Chang, A. Dienes, J. P. Heritage, P. J. Delfyett, S. Dijaili, F. G. Patterson.Femtosecond self- and cross-phase modulation in semiconductor laser amplifiers. J. Select. TopicsQuantum Electron., 2 no. 3:523–539, 1996.

74

Page 75: Modeling of Quantum Dot Semiconductor Optical … of Quantum Dot Semiconductor Optical Ampli ers for Telecommunication Networks vorgelegt von Diplom-Ingenieur Dmitriy Puris von der

[97] A. Mecozzi. Analytical theory of four-wave mixing in semiconductor amplifiers. Opt. Lett., 19, no.12:892–894, 1994.

[98] A. Uskov, J. Mørk, J. Mark, M. C. Tatham, and G. Sherlock. Terahertz four-wave mixing insemiconductor optical amplifiers: Experiment and theory. Appl. Phys. Lett., 65:944 (3 pp.), 1994.

[99] J. Mørk, A. Mecozzi. Theory of nondegenerate four-wave mixing between pulses in a semiconductorwaveguide. IEEE J. Quantum Electron., 33, no. 4:545–555, 1997.

[100] P. Borri, W. Langbein. Four-wave mixing dynamics of excitons in InGaAs self-assembled quantumdots. J. Phys.: Condens. Matter, 19:295201 (20pp), 2007.

[101] T. Akiyama, O. Wada, H. Kuwatsuka, T. Simoyama, Y. Nakata, K. Mukai, M. Sugawara, andH. Ishikawa. Nonlinear processes responsible for nondegenerate four-wave mixing in quantum-dotoptical amplifiers. App. Phys. Lett., 77:1753–1755, 2000.

[102] K. Kikuchi, M. Kakui, C.-E. Zah, T.-P. Lee. Observation of highly nondegenerate four-wavemixing in 1.5 µm traveling-wave semiconductor optical amplifiers and estimation of nonlinear gaincoefficient. IEEE J. Quantum Electron., 28:151–156, 1992.

[103] O. Qasaimeh. Theory of four-wave mixing wavelength conversion in quantum dot semiconductoroptical amplifiers. IEEE J. Quantum Electron., 16:993–995, 2004.

[104] M. Shtaif and G. Eisenstein. Analytical solution of wave mixing between short optical pulses in asemiconductor optical amplifier. Appl. Phys. Lett., 66:1458–1460, 1995.

[105] T. Akiyama, H. Kuwatsuka, N. Hatori, Y. Nakata, H. Ebe, M. Sugawara. Symmetric highly efficient(∼0 dB) wavelength conversion based on four-wave mixing in quantum dot optical amplifiers. IEEEJ. Photonis Technol., 14, no. 8:1139–1141, 2002.

[106] P. Borri, W. Langbein, J. M. Hvam, F. Heinrichsdorff, M.-H. Mao, and D. Bimberg. Ultrafast gaindynamics in InAs-InGaAs quantum-dot amplifiers. IEEE Photon. Technol. Lett., 12, no. 6:594–596,2000.

75