**5.1 Three rate equations model**

QD-SOA characteristics can be studied using REs system constructed from three equations which describe the dynamics in WL, GS and ES of QD layer. The QD inhomogeneity due to

The Composition Effect on the Dynamics of Electrons in Sb-Based QD-SOAs 159

Four REs system is used here to study Sb-based QD-SOAs and is built depending on the energy band diagram considered in Fig. 2. The advantages of using four rate equations system are twofold: 1) including of SCH barrier layer dynamics which is important in practice and 2) this RE model is more appropriate with the use of global Fermi-energy in the gain calculations which is described in section 2. In this four REs system, carriers are assumed to be injected with current density, *J*, in the SCH barrier layer then recombine at a

section 5.1. The processes in the four REs system are represented by the following REs model which covers carrier density, *Ns*, in SCH barrier layer, in addition to: *Nw*, *h* and *f*

> *Ns ssw <sup>J</sup> NNN t qLw s sr we*

2 2

(1 ) (1 ) (1 ) 2 2 21 12

*N N h Nh N wsw w w w N N <sup>t</sup> we s w w wR*

*h NL h NLh ww ww f h f h*

 

<sup>2</sup> ( ) (1 ) (1 ) max (2 1)

*g w f f <sup>h</sup> <sup>f</sup> <sup>h</sup> <sup>f</sup> <sup>p</sup> <sup>c</sup> S f <sup>L</sup> av t n <sup>R</sup> <sup>N</sup> <sup>g</sup> <sup>Q</sup>*

(1 )

 

*tN N Qw Q w* 

to the WL which works as a common reservoir for the

 

. Other processes are described in

(11)

(14)

(12)

(13)

 

Fig. 2. Energy diagram of QD-SOA system.

, relax at rate (1 / )*<sup>s</sup>*

which are covered also in the three REs model

21 12 1

 

carriers. The escape rate from WL to the SCH is (1 / ) *we*

**5.2 Four rate equations model** 

rate (1 / ) *sr* 

fabrication imperfections is introduced through gain as discussed in section 3 above. Examples of three REs system can be found in [11, 14]. In the three REs system, carriers are assumed to be injected with current density, *J*, in WL where the barrier layer effect is neglected. In WL, carriers are recombining at a rate (1 / ) *wR* , relax at a rate 2 (1 / ) *<sup>w</sup>* to the ES. ES relaxes quickly to GS at rate 21 (1 / ) , thus it does not contribute to gain. The escape component from ES to WL is 2 (1 / ) *<sup>w</sup>* and from GS to ES is 12 (1 / ) . Both GS and ES are assumed to be recombining at the same rate 1 (1 / ) *<sup>R</sup>* , see Fig. 2. The Pauli blockings (1 ) *h* and (1 ) *f* are taken for the nonempty ES and GS, respectively. For example, if ES is empty, then (*h=0*) and carriers can occupy the state, if it is fully occupied, then (*h=1*) and no further carriers can occupy it. The above described processes are represented by the following REs model for the carrier density *Nw* in the WL, occupation probabilities *h* and *f* in the ES and GS, respectively

$$\frac{\partial N\_{\text{uv}}}{\partial t} = \frac{I}{qL\_{\text{uv}}} - \frac{N\_{\text{uv}}(1-h)}{\tau\_{\text{uv}}\mathbf{2}} + \frac{N\_{\text{uv}}h}{\tau\_{\text{2}\text{uv}}} - \frac{N\_{\text{uv}}}{\tau\_{\text{uv}\mathbf{R}}} \tag{6}$$

$$\frac{\partial h}{\partial t} = \frac{N\_{\text{UV}}L\_{\text{UV}}(1-h)}{N\_{\text{Q}}\sigma\_{\text{W}}\mathbf{2}} - \frac{N\_{\text{UV}}L\_{\text{UV}}h}{N\_{\text{Q}}\sigma\_{\text{Z}}\mathbf{2}w} - \frac{(1-f)h}{\tau\mathbf{2}\mathbf{1}} + \frac{f(1-h)}{\tau\mathbf{1}\mathbf{2}} \tag{7}$$

$$\frac{\partial f}{\partial t} = \frac{(1-f)h}{\tau\_{21}} - \frac{f(1-h)}{\tau\_{12}} - \frac{f^2}{\tau\_{1R}} - S\_{av} \Gamma \frac{g\_{\text{max}} \text{(hw}\_p)}{N\_Q} (2f-1)L \left. \frac{c}{n\_g} \right| \tag{8}$$

where *q* is the electron charge, *Lw* is the effective thickness of the active layer, *Г* is the optical confinement factor and *NQ* is the surface density of QDs. The average signal photon density *Sav* is given by [14]

$$\mathbf{S}\_{\text{av}} = \frac{\mathbf{g}\_s P\_{\text{in}} \mathbf{n}\_{\text{g}}}{\hbar w\_p L\_{\text{w}} D L \mathbf{g}\_{\text{max}} c} \tag{9}$$

*Pin* is the input signal power to the SOA, *ng* is the group refractive index, *ћ* is the normalized Plank's constant, *wp* is the peak frequency, *D* is the strip width, *L* is the cavity length, *c* is the free space light speed, *gmax* is the peak material gain taken at the peak frequency. The input signal wavelength is assumed to be injected to the QD SOA at a peak wavelength of GS for each structure. The single-pass gain of the structure is given by [11]

$$\mathcal{g}\_{\mathcal{S}} = \exp\left[ (\mathcal{g}\_{\text{max}} \Gamma - a\_{\text{int}})L \right] \tag{10}$$

where *αint* is the loss coefficient. Although WL is assumed here to receive carriers by current injection or by escapes from QD ES, but the three RE models consider it as a common reservoir [10, 15]. From the architecture of these three RE model, ES works as a common reservoir since it receives carriers from the states above and below it (WL and GS, respectively). So, ES is referred to as a reservoir for GS [8]. This is contradicted with the experimentally evidenced that carriers stay long in WL and short in ES [16]. Due to this and other reasons, discussed below, four REs system is more appropriate.

Fig. 2. Energy diagram of QD-SOA system.
