Quantum-Dot Semiconductor Optical Amplifiers : State Space Model versus Rate Equation Model

A simple and accurate dynamic model for QD-SOAs is proposed. The proposed model is based on the state space theory, where by eliminating the distance dependence of the rate equation model of the QD-SOA; we derive a state space model for the device. A comparison is made between the rate equation model and the state space model under both steady state and transient regimes. Simulation results demonstrate that the derived state spacemodel not only is much simpler and faster than the rate equationmodel, but also it is as accurate as the rate equation model.


Introduction
During the last decade, the potential capabilities of QD-SOAs for use in all-optical signal processing and optical communication systems have been intensively studied.Among these capabilities, ultrafast gain recovery [1][2][3][4][5], high saturated output power [6,7], pattern-effect free signal amplification at high speeds up to 80 Gb/s [8][9][10], pattern-effect free XGMbased wavelength conversion at 160 Gb/s [11], capability of operation at Tb/s speeds in presence of a control signal [12], amplification of high bit rate multichannel signals [13,14], low noise figure [15], small dimensions, and integration with other optoelectronic devices such as laser diodes and optical modulators have great importance in any optoelectronic system.
In recent years, several models have been proposed for describing the electrical and optical characteristics of QD-SOAs.Among these theoretical models, the most accurate models are based on semiconductor Maxwell-Bloch equations [16][17][18][19][20].However, the numerical calculations associated with this model are extremely time-consuming and require huge amount of memory.A simplified approach to model QD-SOAs which is known as rate equation model (REM), has demonstrated an excellent agreement with experimental results [4,21].The REM includes a set of coupled differential equations to give details of carrier dynamics and optical properties of QD-SOA.To include the carrier dynamics in the REM, in some papers the electron-hole pairs are considered as exciton and only the carrier dynamics in the conduction band (CB) is taken into account [22][23][24][25].In some other papers, the holes dynamics is included by using quasi-Fermi level in the valence band (VB) [26].Also, the dynamics of electron and hole are considered separately in some articles [4,[27][28][29][30][31].This model is known as "electron-hole model" [30], where the rate equations for electrons and holes are written separately.In this paper, we have employed the last approach to give details of the investigated QD-SOA [29,31].
Although the REM of QD-SOA is much simpler and faster than QD Maxwell-Bloch equations, because of time as well as distance dependence of the rate equations, the long computation time is still a big concern in some applications.Furthermore, since the simulation run time is very sensitive to the distance between two adjacent nodes in the spaced distance-time grid, a huge memory may be required during the simulation execution, especially in condition that a long sequence of optical pulses have to be considered in numerical simulations.Therefore, our objective is to derive a simple and efficient dynamical model for QD-SOA that calculates the optical gain and output of the device with an acceptable precision.For this purpose, the average values of occupation probabilities along the QD-SOA cavity are considered as the variables of the system and the distance dependence of the REM is eliminated.In the state space model, the QD-SOA is considered as a black box; that is, it is viewed mainly in terms of its input and output characteristics.
This paper is organized as follows.In Section 2, the physical structure of the investigated QD-SOA as well as the rate equations of the device are presented.In Section 3, we derive a SSM for QD-SOA.In Section 4, the effects of the homogeneous and inhomogeneous broadenings are included in the State Space Model of QD-SOAs.Comparison between the results obtained from SSM with results of the REM is presented in Section 5. Finally, Section 6 gives a summary of our work.

Physical Model and Theory
The investigated device is an InAs/GaAs QD-SOA that operates around 1.3 m.The active region of the device consists of seven stacked layers of self-assembled InAs QDs which are grown on a GaAs substrate (see Figure 1).
The detailed band structure of this QD-SOA is illustrated in Figure 2. The self-assembled QDs have 3 nondegenerate energy levels in the CB and 8 nondegenerate energy levels in the VB, and are accompanied by two dimensional wetting layer (WL) states [29].
The dynamic behavior of the QD-SOA is determined by photon as well as carrier rate equations.
The rate equations of the investigated QD-SOA can be found in [29,31] and are summarized as follows.
Photon rate equation: where where

Electron States Rate Equations.
For the GS: For the th ES ( = 1, 2): For the WL state: where For the th ES ( = 1, 2): For the th ES (3 ≤  ≤ 7): For the WL state: where  and  are independent variables while all other variables, that is, occupation probabilities, are dependent variables of the REM.Details of the REM and also simulations parameters of the investigated QD-SOA can be found in [31].
The rate equations can be solved for a given injected current and input photon density.By solving the rate equations, the longitudinal and temporal variations of the photon density and occupation probabilities can be determined.

SSM of QD-SOA
The REM of the investigated QD-SOA was presented in Section 2. As stated before, our objective is to develop a simple and accurate SSM for QD-SOAs.In order to derive the SSM of the QD-SOA, the dependency of the occupation probabilities on distance must be eliminated.By considering the average values of the occupation probabilities as the REM variables, the carrier density along the cavity gets a uniform distribution and consequently, -independent variables will be appeared in the REM of the QD-SOA.By this assumption and after some manipulations, the SSM of the QD-SOA is derived, where the average values of the occupation probabilities are the state variables of the system.To find out whether the average values of the occupation probabilities are the appropriate state variables of the SSM, we need to know the effects of uniformity of the occupation probabilities along the cavity on the final results of the REM.
Our calculations demonstrate that the variance of the carrier distribution along the cavity does not have a significant impact on the gain dynamics of the amplifier [31].In other words, the output is weakly dependent on the carrier density variations along the cavity.Consequently, it is reasonable to assume that in the state space model of the QD-SOA, the carrier density distribution along the QD-SOA cavity is completely uniform; that is, we assume that the variance of the carrier density distribution along the cavity is zero.By this assumption, the average values of the occupation probabilities become the state variables of the device [32].In the following, the details of the mathematical derivation of the SSM are presented.We denote the distance independent occupation probabilities as    (),    (),   , and   .These quantities are the average values of occupation probabilities along the cavity and are considered as the state variables of the QD-SOA in the state space.Therefore, the modal gain of QD-SOA in the SSM takes the following form  QD (), which we name it as "average modal gain, " and is given by Since the state variables do not fluctuate along the QD-SOA cavity, the average modal gain will also be unchanged along the cavity.Depending on the supposed model for QDs, the number of state variables is determined.In our model, InAs QDs have totally 11 energy levels.Considering the WL states, the dynamics of our investigated QD-SOA can be described by 13 state variables.As we know from state space theory, the state update equations of a nonlinear system are generally given by  where () and () denote, respectively, the state vector and input vector of the system.Also, the output relation of this system is given by where  1 and  2 are nonlinear functions of  as well as () and (), and where () denotes the output vector.To derive an SSM for QD-SOA, we begin from photon rate equation and obtain a closed form relation between input and output photon densities.This relation enables us to derive a new relation for stimulated emission rate, and therefore in the next step we rewrite the rate equations of QD-SOA in the form of state update equations.Derivation details are as follows.
The photon rate equation of the QD-SOA can be rewritten using the state variables as follows: By integrating (18) we have Therefore, the relation between the input and output photon densities of QD-SOA becomes where  in and  out are, respectively, the photon densities at the input facet ( = 0) and at the output facet ( = ) of the QD-SOA, and  is the cavity length.On the other hand, from (18) we have Substituting ( 21) in ( 9), one can obtain Integrating ( 22) from 0 to  and averaging along the cavity, one can get the following expression for stimulated emission rate for the th energy state as where (, ) =  out (), (0, ) =  in ().Substituting ( 20) in (23), we obtain the relation of stimulated emission rate as which we name it as "effective" stimulated emission rate.Also, the effective spontaneous emission is given by Similarly, the effective electron capture and escape rates are, respectively, given by  The effective capture and escape rates for holes in the VB levels are similar to ( 26) and ( 27), except that the superscript  is replaced by .Since the rate equations and consequently the state update equations are not explicit functions of time, QD-SOA is an autonomous system.Hence, the SSM of the device takes the following form: where ]  is the state vector of the system, () =  in () is the input variable, () =  out () is the output variable,  1 is the resultant rate of carriers transition rates,  2 describes the relation between the input and output photon densities, that is, (20).Using (24) to (27), we develop the SSM of QD-SOA in the form of (27).The state update equations are given by the following differential equations.

Electron Levels.
For the GS: For the th excited state, where  = 1, 2: And for the WL state: 3.2.Hole Levels.For the GS: For the th excited state, where  = 1, 2: And for the WL state: The output relation of the SSM is described in (20).

Including the Homogeneous and Inhomogeneous in the State Space Model of QD-SOAs
In this section, we develop our proposed method for a more comprehensive REM of QD-SOAs.The REM which is used in this paper is proposed by Kim et al. [4].The investigated device is an InAs/GaAs QD-SOA which operates around 1.3 m.Since QDs are grown self-assembly, they have slightly different properties induced by size fluctuations.It is assumed that self-assembled QDs have a size distribution with a Gaussian profile and consist of 181 spectral groups, where QDs in each QD group are assumed identical.The separation between the transition energies of two adjacent groups is considered Δ = 1 meV.Also, it is assumed that all QDs are spatially isolated and each QD can exchange only with a common carrier reservoir.Details of the physical structure and energy band diagram can be found in [5,33].The dynamics of each QD group is described by six state variables [33].Thus, the QD ensemble is described by 1086 state variables (6 × 181).Including the QW state in the CB and VB, the investigated QD-SOA is described by an NSSM with 1088 state variables.The state update equations of the QD-SOA are summarized as follows [33]: Advances in OptoElectronics 7 In ( 37)-( 40),  (V) , ,  (V) , ,  (V) , , and  (V)  are the state variables of the system.For instance,  (V) , signifies the state variable associated to the ground state of the th QD group in the conduction (valence) band.Details of this SSM as well as simulation results related to this model can be found in [33].

Simulation Results
To investigate the accuracy of the SSM, in this section we compare the simulation results of the SSM with those of the REM.We use the finite difference method (FDM) to solve the REM and SSM [31].To solve the REM using FDM, the cavity length is partitioned into M longitudinal sections (see Figure 3), where the physical quantities in each section are assumed to be constant along the section [31].To simulate the response of the QD-SOA using the REM and SSM, we need to know the initial state of the system.The initial state (occupation probabilities at  = 0) can be found by solving the REM and SSM at steady state condition for  in = 0. Since at steady state the time derivatives are zero, the initial state of the system is equivalent to finding the roots of a system of 13 coupled nonlinear algebraic equations, which can be done by Newton method.
In order to compare the dynamic response of the SSM with that of the REM, the time response of both models are evaluated in condition that a Gaussian shaped pulse with a width of 0.4 ps and 3 pJ energy is applied at the input facet (see Figures 4(a Since the accuracy of the results of the REM depends on the discretization errors, the rate equations of the QD-SOA are solved for different values of Δ.As seen from the figure, with enhancement the accuracy of numerical calculations, that is, considering smaller values for Δ, the results of the REM become closer to the results obtained by the SSM.Also, Figure 4(c) illustrates the gain saturation curve of the QD-SOA under different values of Δ. Figure 4(c) demonstrates that the SSM of the QD-SOA not only is accurate in the linear region (low input power), but also is accurate at the nonlinear region (under the gain saturation).For Δ = /4000, the results of the REM are very close to the results obtained from the SSM, where the percentage error in calculating the optical gain and output signal is less than 0.5%.These results imply that with lowering the discretization errors, these two models represent even closer results.As is evident, by dividing the cavity length into smaller longitudinal sections the accuracy of the FDM calculations increases.Since the simulation run time of the REMs is very sensitive to the distance steps in the distance-time grid, considering smaller values for Δ increases the computation time.The required time for solving the REM is much longer than the simulation run time of the SSM.Furthermore, while the stability of the numerical solving of the state update equations is guaranteed by choosing a small value for Δ, stable solving of the rate equations is dependent on appropriate selection of both Δ and Δ.Therefore, the convergence problems associated with solving the SSM are reduced compared to REMs.

Conclusion
A state space model for a typical QD-SOA was proposed.For this purpose, the theoretical basis of the SSM of the QD-SOA was developed, where it was shown that the gain of the QD-SOAs is a weak function of distribution of carriers along the cavity.Based on this result, the average values of the occupation probabilities along the QD-SOA cavity were considered as the state variables of the system.Consequently, we derived an SSM for the QD-SOA.We carried out a comparison between the results of the derived SSM and the rate equation model, where both the static and dynamic responses of the QD-SOA were compared with results obtained from the REM.Simulation results showed that the SSM of the QD-SOA is accurate in the linear as well as nonlinear region.Numerical calculations demonstrated that not only the SSM is simpler and faster than the rate equations model, but also the convergence problems associated with solving QD-SOA rate equations are decreased.

Figure 1 :
Figure1: Schematic representation of QD-SOA with dots-in-a-well (DWELL) structure, where a sevenfold-stacked self-assembled InAs QD active region is sandwiched between two AlGaAs cladding layers.InAs QDs are covered with a 5 nm thick InGaAs capping layer.The QD layers are separated by 33 nm thick GaAs spacer layers which are considered for strain relaxation.

Figure 2 :
Figure 2: The energy band diagram of the active region of the QD-SOA.The energy separations of the QD electron and hole states are 70 meV and 10 meV, respectively.The radiative transitions are shown in the figure.

Figure 3 :
Figure 3: Schematic diagram of the QD-SOA cavity which is divided into M identical sections to be used by FDM, where  =  ⋅ Δ.