## Abstract

Quantum dot-semiconductor optical amplifiers (QD-SOA) attracted strong interest for applications in optical communications and in all-optical signal processing due to their high operation rate, strong nonlinearity, small gain recovery time of about few picoseconds, broadband gain, low injection current and low noise figure (NF). In this chapter, we present the theoretical investigation of the gain recovery time acceleration in DQ SOA; the specific features of the cross gain modulation (XGM) in QD-SOA; the influence of the optical injection on the dynamics of QD-SOA based on the QD in a well (QDWELL) structure. We describe the following applications of QD-SOA: the all-optical ultra-wideband (UWB) pulse generation based on the Mach-Zehnder interferometer (MZI) with a QD-SOA; the ultra-fast all-optical signal processor based on QD-SOA-MZI; the ultra-fast all-optical memory based on QD-SOA. The contents of the chapter are mainly based on the original results.

### Keywords

- quantum dot
- semiconductor optical amplifier
- cross-gain modulation
- all-optical processor
- all-optical radio signal generation

## 1. Introduction

Quantum dot-semiconductor optical amplifiers (QD-SOA) are characterized by ultrafast gain recovery time (GRT) of the order of magnitude of several picoseconds, broadband gain, low noise figure (NS), high saturation output power, and high four-wave mixing (FWM) efficiency [1]. QD-SOA can be used as ultra-wideband (UWB) polarization-insensitive high-power amplifiers, high-speed signal regenerators, and wideband wavelength converters (WC) [1, 2, 3, 4, 5]. These unique features of QD-SOA are essentially due to the concentration of the injected electrons and holes into nanosized QD [1]. QD is a nanostructure characterized by the electron and hole confinement in all three dimensions [2, 6]. QD is a cluster with the dimensions of several nanometers made of a semiconductor material [2]. In a QD charge carriers occupy a limited number of energy levels similarly to an atom, and the density of states is quantized [2]. As a result, QD has freedom of a wavelength choice [1]. A QD contains hundreds of thousands of atoms [2]. Typically, III-V QD are epitaxially grown on a semiconductor substrate [2]. The process of the spontaneous formation of three-dimensional islands during strained layer epitaxial growth is known as the Stranski-Krastanov mechanism [2]. A continuous film of a quantum well (QW) thickness lies underneath QD, and it is called the wetting layer (WL) [2]. The lattice constant of the deposited semiconductor material must be larger than that of the substrate [2]. For instance, an InAs film with the lattice constant of 6.06 Å can be deposited on a GaAs substrate with the lattice constant of 5.64 Å or on an InP substrate with the lattice constant of 5.87 Å [2]. Stranski-Krastanov grown QD typically has a pyramidal shape with a base of 15–20 nm and a height of about 5 nm; the QD density per unit area is between 10^{9} and 10^{12} cm^{−2} [1, 2]. The methods of the QD energy levels and density of states evaluation are presented in Ref. [6].

The theory of the optical signal amplification and processing based on the density matrix equations for the electron-light interaction and the optical pulse propagation equations has been developed [7]. The QD spatial localization, the inhomogeneous spectral broadening caused by the QD size, shape, composition, and stain distribution, the carrier capture from WL, the carrier emission to WL, intradot population relaxation, and homogeneous spectral broadening have been taken into account, and the nonlinear optical response has been also investigated [7]. The phenomenological approach to the QD-SOA theory is based on the system of rate equations for photons and charge carriers in QD [2, 5, 8]. In such a case, the QD is considered as a three-state system including the ground state (GS), the excited state (ES) and WL [2, 5, 8]. The electron dynamics in QD is assumed to be slower than the hole dynamics, and for this reason, only the rate equations for electron populations of GS, ES and WL are included in the dynamic model [2, 5, 8]. Using this model, we investigated theoretically acceleration of gain recovery and dynamics of electrons in QD-SOA [9, 10]. We have shown that the QD-SOA GRT may be substantially decreased, and the patterning effect is reduced by increasing the optical pump power, while the chirp in QD-SOA is about one order of magnitude lower than the one in the bulk SOA [9, 10]. We studied theoretically the cross-gain modulation (XGM) process in QD-SOA taking into account the QD-SOA inhomogeneous spectral broadening [11]. We have shown that XGM in QD-SOA occurs at larger detuning between the pump and signal light waves as compared to the bulk SOA, the asymmetric chirp may be diminished by the bias current increase, and XGM process slows down in the nonlinear regime [11].

Recently, QD-SOA and lasers based on a novel quantum dot-in-a-well (QDWELL) structure have been proposed where the self-assembled QD has been grown into QW with the discrete energy levels and the two-dimensional (2D) electron gas instead of ordinary QD laser and SOA with the continuous carrier energy in WL [12, 13]. The operation of QD-SOA based on a QDWELL structure (QDWELL SOA) has been investigated both theoretically and experimentally [12, 13, 14, 15, 16]. The complicated dynamics of QDWELL SOA is described by the system of the rate equations for the electrons and holes in QD and QW including the strongly nonlinear electron and hole scattering rates for the carrier scattering between QD and QW [12, 17]. The operation rate of QDWELL SOA is limited by the desynchronized recovery dynamics of electrons and holes caused by the different microscopic scattering rates [15]. We have shown theoretically that the electron and hole dynamics in QDWELL lasers and SOA can be synchronized by a sufficiently strong optical injection and consequently the QDWELL lasers and SOA performance including the operation rate can be significantly improved [18, 19, 20, 21, 22].

Optical signal processing is based on the linear and nonlinear optical techniques for the digital, analogous and quantum information [23]. It is a promising technology for increasing the processing speed of devices, the capacity of optical links, and reducing of energy consumption [23]. In particular, QD-SOA are excellent candidates for high-speed data and telecommunication applications due to their ultrafast gain dynamics and pattern effect-free amplification [15]. We proposed the following novel applications of QD-SOA in optical communications:

a novel all-optical method of the impulse radio ultra-wideband (IR-UWB) pulse generation based in an integrated Mach-Zehnder interferometer (MZI) with QD-SOA as an active element inserted into one arm of the integrated MZI [24, 25, 26];

an ultra-fast all-optical processor based on QD-SOA [27, 28]; and

The detailed results of the numerical simulations are presented in Refs. [9, 10, 11, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30]. In this work, we discuss the theoretical models of QD-SOA, QDWELL SOA, and some devices for the all-optical signal processing, summarize the results obtained and present the necessary numerical estimations. The chapter is organized as follows. In Section 2, we discuss the dynamics of the gain recovery and XGM processes in QD-SOA. In Section 3, we consider the influence of the optical injection on the QDWELL SOA performance. The applications of QD-SOA in optical signal processing are discussed in Section 4. The conclusions are presented in Section 5.

## 2. Specific features of the QD-SOA dynamics

### 2.1. Theoretical model of QD-SOA

In this section, we theoretically investigate the gain recovery process and XGM in QD-SOA. The analysis is based on the simultaneous solution of the rate equations for the electron density per unit volume in WL

The carrier transitions between WL, ES and GS are characterized by different relaxation times. The fast transition processes between WL and ES, and between ES and GS are described by the relaxation times

The equations for

Here,

where

where

The cross-section

where

In our analysis of the QD-SOA dynamics, we are interested in the temporal dependence of the power

Integrating Eq. (5), we obtain the phases

The chirp

System of Eqs. (1)–(3) with the average pump and signal photon densities (10) describes the gain recovery and XGM processes in QD-SOA. These equations are strongly nonlinear, and for this reason, they are extremely complicated. Their analytical solution in a closed form is hardly possible. We solved the system of Eqs. (1)–(5) numerically for the following typical values of the material parameters:

### 2.2. Theoretical analysis of the gain recovery process in QD-SOA

The QD-SOA performance is mainly determined by their GRT and the magnitude of the chirp

Consider now the influence of the bias current and the CW probe optical wave intensity on the QD-SOA dynamics in two cases of the optical signal wave with the power of

We start with the analysis of the QD-SOA dynamics in the absence of the probe optical wave. In the PRBS case, the simulation results show that at a low bias current of an order of magnitude of

### 2.3. Peculiarities of XGM in QD-SOA

Consider now the peculiarities of XGM in QD-SOA caused by the gain inhomogeneous broadening which is due to the variation of QD size, shape, and local strain [11]. XGM is an essentially nonlinear process in SOA caused by the carrier density change influence on the input signal waves [36]. Typically, XGM in SOA may realize the wavelength conversion [36]. The XGM process in QD-SOA involves three types of transitions: (1) fast transitions between ES and GS characterized by

We used the bias current values

Consider now the temporal dependence of the input signal power

Consider the possibility of XGM in QD-SOA between the waves with the detuning larger than the QD-SOA homogeneous broadening. In such a case, we divide QD into groups with different resonant frequencies caused by the inhomogeneous broadening. This time, we solve Eqs. (1)–(3) numerically for the QD group 1 and 2 with a detuning substantially larger than the QD-SOA homogeneous broadening [11]. The numerical simulation results show that the strong interaction through WL for these QD groups occurs for a comparatively low bit rate of 10 Gb/s and significantly diminishes at 40 Gb/s. At the large bit rates, the ES and GS populations of the first QD group do not follow the changes of the ES and GS populations of the second QD group, the WL electron concentration is slightly varying, and the XGM effect vanishes. The output optical power

The comparison of the performance of bulk, multiquantum well (MQW) and QD-SOA shows that XGM can be realized in the bulk SOA for the bias current of

## 3. The influence of the optical injection on the QDWELL SOA performance

In this section, we consider XGM in the TW QDWELL SOA mainly following references [20, 21, 22].

The block diagram of the TW QDWELL SOA is presented in Figure 3. Typically, a QDWELL active region structure consists of 10–15 InGaAs quantum well (QW) layers with a height of about 4 nm containing embedded InAs QD with a size of approximately 4 nm x 18 nm x 18 nm [12]. The layers of InGaAs/InAS QD are separated by 33 nm GaAS spacers providing the strain relaxation between successive QD layers [13]. The electric bias current is injected into the QW layers which represent the reservoir of the 2D carrier gas unlike the WL with the continuous energy of carriers in ordinary QD-SOA [12]. The ridge width and the waveguide length are

For the moderate bias currents and comparatively slow processes, the dynamics of the ES-GS transitions can be adiabatically eliminated [12]. Then, the LS rate equations have the form [12, 17]:

where

The equations for the pumping and signal wave photon density per unit area

The photon density per unit area

where

The phases

The relation between the pumping and signal wave optical power

where

We solved numerically the system of Eqs. (13)–(16) for the typical values of the material parameters

We evaluated the QDWELL SOA gain for

Consider now the synchronization of the carrier dynamics and gain recovery process for the super-Gaussian pulse of the signal wave

Consider now the QDWELL SOA large signal response for the electrical PRBS signal with the length of

The QW and QD carrier behavior changes drastically in the case of the strong optical pumping

Consider now the influence of the optical pumping on the XGM and cross-phase modulation (XPM) of the co-propagating pumping and signal optical waves in QDWELL SOA [20, 21, 22]. We suppose that these waves have the same polarization corresponding to a maximum value of the gain since the gain in QDWELL SOA has strong polarization dependence [4]. We investigated the XGM in the in QDWELL SOA both in the pulse regime and in the PRBS regime.

In the pulse regime, for the pulse duration of

The synchronized dynamics of the QW and QD carriers in the PRBS regime due to the high level of the optical power also provides the efficient XGM and eliminates the pattern effect [21]. The chirp of output signal wave

We theoretically investigated the extinction ratio (ER) dependence on the CW pumping power

## 4. The applications of QD-SOA in optical signal processing

In this section, we discuss the applications of QD-SOA in all-optical generation of UWB impulse radio signals [24, 25], ultra-fast all-optical processor [27, 28] and all-optical memory [29, 30].

### 4.1. All-optical generation of UWB impulse radio signals based on MZI with QD-SOA

UWB communication systems are characterized by low power consumption, immunity to multipath fading, precise object location, and high data rates [39]. However, they can operate in the frequency range from 3.1 to 10.6 GHz with an effective isotropic radiated power level of less than -41 dBm/MHz according to the U.S. Federal Communication Commission (FCC) decision [24, 40]. For this reason, UWB wireless systems are limited by short distances of a several tens of meters [40]. In order to increase the area of coverage, the UWB- over-optical-fiber (UROOF) technology had been proposed [24, 39, 40]. UWB-over-fiber technology can be used for radars, wideband wireless personal area networks (WPAN), sensor networks, imaging systems UWB positioning systems, and so on (see [24] and references therein). In particular, impulse-radio (IR) UWB technology is important where the information is carried by a set of narrow electromagnetic pulses with a bandwidth inversely proportional to the pulse width [24]. Carrier-free impulse modulation avoids complicated frequency mixer, intermediate frequency carrier and filter circuits and has better pass-through characteristic due to the base-band transmission [39]. For instance, Gaussian monocycle and doublet pulses, which are the first and second order derivatives of Gaussian pulse, respectively have lower bit error rate better multipath performance and wider bandwidth as compared to other impulse signals [39, 40].

There exist different methods of the optical IR UWB generation [39, 40, 41, 42, 43, 44]. Different optically based systems for the generation of the Gaussian IR UWB monocycles and doublets may include an electro-optic phase modulator (EOPM), a single mode fiber (SMF), erbium-doped fiber amplifier (EDFA), SOA, a fiber Bragg grating (FBG), a photodetector (PD), a Sagnac interferometer, photonic microwave filters [39, 40, 41, 42, 43, 44]. The shortages of such systems are the necessity of the complicate electronic circuit for the generation of the short electric Gaussian pulses, the use of EOM and long SMF.

We proposed a novel all-optical method of IR UWB pulse generation based on the integrated MZI with QD-SOA inserted into one arm of the integrated MZI [24]. An intensity-dependent signal interference occurs at the output of the MZI with QD-SOA. The proposed UWB IR signal generation process is based on the XGM and XPM processes in QD-SOA, which are characterized by strong optical nonlinearity and high operation rate as it has been mentioned earlier.

The block diagram of the proposed all-optical UWB IR signal consisting of a CW laser, MZI with a QD-SOA as an active element in the upper arm of the MZI, and a pulsed laser is shown in Figure 5. A CW signal of a wavelength

where

The spectrum of the simulated UWB IR signal exhibits the filtering features of the proposed generator [24]. Indeed, for the Gaussian pulses duration

### 4.2. Ultrafast all-optical processor based on QD-SOA

The major application areas of SOA-based MZI are all-optical logic gates, optical WC, and optical regenerators. The latter devices provide the so-called 3R regeneration [3], that is, re-amplification, re-shaping, and re-timing which are necessary for the elimination of the noise, crosstalk and nonlinear distortions and for the transmission of the good quality signals over sufficiently large distances in all-optical networks (see [27, 28] and references therein).

We proposed a theoretical model of an ultrafast all-optical signal processor based on MZI containing QD-SOA in each arm (QD-SOA-MZI) where XOR logic operation, wavelength conversion, and 3R signal regeneration can be realized simultaneously by AO-XOR logic gates for the bit rates up to

The block diagram of AO-XOR logic gate based on the QD-SOA-MZI is shown in Figure 7.

The theoretical analysis of the proposed processor is based on the QD-SOA dynamics described by Eqs. (1)–(5) and the expression for the MZI output power (21). However, this time, QD-SOA are situated in both arms of the MZI, and the QD-SOA amplification factors

For the typical values of LEF

We start with the operation of the logic gate based on the QD-SOA-MZI, which consists of a symmetrical MZI with one QD-SOA in each arm [27, 28]. Two optical control beams

Suppose that the data stream at the input of the QD-SOA-MZI is absent:

Consider now the wavelength conversion in the proposed processor. An ideal WC must be characterized by the following properties: transparency to different bit rates and signal formats, fast setup of the output wavelength, the possibility of conversion to shorter and longer wavelengths, moderate input power levels, the possibility for no conversion regime, insensitivity to the input signal polarization, the output signal with low chirp, high ER and large signal-to-noise ratio (SNR), and simple implementation [45]. SOA is a promising candidate for applications in WC because it possesses these characteristics.

WC performance can be substantially improved by replacing the bulk SOA with QD-SOA due to its specific features discussed in Section 2: the pattern-free high-speed wavelength conversion of optical signals by XGM, a low threshold current density, a high material gain, high saturation power, broad gain bandwidth, and a weak temperature dependence as compared to bulk and MQW SOA [9, 10, 11]. In the proposed ultrafast all-optical processor, the advantages of the QD-SOA as a nonlinear component and of the MZI as a system with the controlled output signal [27, 28]. Consider the situation where one of the inputs signals

The deterioration of the ultrafast all-optical processor with the bit rate increase is shown in Figures 8 and 9. For the bias current

The proposed ultrafast all-optical processor can solve three problems of the short pulse 3R regeneration mentioned above [27, 28]. The efficient pattern-effect-free signal re-amplification can be realized in each of the QD-SOA-MZI by the corresponding QD-SOA. The wavelength conversion based on all-optical logic gate discussed above can provide the reshaping since only the data signals can close the gate, while the comparatively weak noise cannot close the gate. The re-timing in the QD-SOA-MZI-based processor is provided by the optical clock stream, which is also necessary for the re-shaping. If the CW input signal is replaced with the optical clock stream, the 3R regeneration can be carried out simultaneously with the logic operations [27, 28].

### 4.3. Ultrafast all-optical memory based on QD-SOA

Optically assisted signal processing combines optical and electronic components [46]. For instance, optical components characterized by high operation rate can carry out some functions very fast, while the electronic components can realize the complex computations using buffers and memory [46]. In optical networks, the bandwidth mismatch between optical transmission and electronic routers requires a different optical signal processing and the study of the optical packet switching (OPS) [47]. The packet switching is used when it is necessary to select a packet of tens or hundreds of bits from a bit stream [48]. The flip-flop memory is an essential component of the packet switch in OPS networks, which is necessary for avoiding the packet collisions during packet routing [48, 49]. Usually, this memory consists of two coupled lasers switching the output signal between the wavelengths

The novel all-optical memory loop is characterized by the following advantages: (1) it includes only one QD-SOA-MZI reducing the complexity of the electronic synchronization scheme; (2) it can operate at the high bit rates up to

The QD-SOA dynamics and the ultra-fast all-optical processor operation principle have been discussed in detail in Section 2 and subsection 4.2, respectively. The theoretical model of the proposed all-optical memory loop is based on the QD-SOA carrier rate Eqs. (1)–(3), the average pump and signal wave photon densities

We solved numerically Eqs. (1)–(3), (10), (11), (21), and (22) for the typical values of the material parameters presented in Section 2 in two cases: (1) the on–off keying (OOK) modulation format, the loop length

For instance, the eye diagram for the 4 PAM modulation format and a bit rate of

The numerical estimations show that for the loop length

## 5. Conclusions

We studied theoretically the dynamics of QD-SOA. We solved numerically the QD-SOA carrier rate equations simultaneously with the truncated equations for the light wave photon density and phase. We have shown that the injection of the additional light wave drastically changes the QD-SOA dynamics. As a result, GRT may be strongly reduced by using the short Gaussian pulses and strong additional light wave. The lower GRT limit is defined by the QD-SOA fastest transition between ES and GS with the transition time

The novel generation of QD lasers and SOA is based on the QDWELL structures where the self-assembled QD is imbedded into QW layers. The QW layers play a role of the reservoir for the 2D carrier gas instead of WL with the continuous carrier energy in ordinary QD-SOA. The complicated QDWELL laser and SOA dynamics are described by the system of LS rate equations for electrons and holes in QD and QW. The QDWELL laser and SOA performance are limited by the strongly desynchronized dynamics of electrons and holes in QD and QW. The different relaxation times of the electrons and holes in QW and QD are determined by the strongly nonlinear electron and hole scattering rates in and out of QD.

We solved numerically the modified LS rate equations and the truncated equations for the pumping and signal waves in the QDWELL SOA for the pulse regime and the large PRBS signal. We have shown that the strong optical injection with the power

The QD-SOA can be successfully used in the all-optical signal processing due to their strong nonlinearity, high operation rate, and low bias current. We proposed a novel all-optical method of the UWB IR signal generation based on the integrated MZI with QD-SOA in each arm as a nonlinear element. The proposed method does not need SMF, EOPM, and FBG reducing complexity and cost of the UWB IR generator. The UWB IR signal generated by the QD-SOA-based MZI has a form of the Gaussian doublet.

We proposed a theoretical model of the ultrafast all-optical signal processor also based on QD-SOA-MZI structure. We have shown theoretically that such a processor may realize logic gate XOR operation, wavelength conversion, and 3R regeneration of the moderately distorted optical signals. The operation of the processor is based on the XGM and XPM in QD-SOA in both arms of the MZI. The processor limiting bit rate depends on the bias current

We proposed a novel architecture of the ultra-fast all-optical memory based on MZI with two QD-SOAs. The numerical simulation results for the OOK and 4 PAM modulation format show that the proposed memory is characterized by high operation rate up to