**5.2 Four rate equations model**

158 Selected Topics on Optical Amplifiers in Present Scenario

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

and from GS to ES is 12 (1 / )

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

> (1 ) 2 2

 

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

*N N h Nh N w w ww <sup>J</sup> t qLw w w wR* 

*h NL h NLh ww ww f hf h*

 

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

*f fh f h f g w <sup>c</sup> <sup>S</sup> f L av t n <sup>R</sup> NQ*

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

*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

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

 *gPn s in g Sav w L DLg c p w*

*tN N Qw Q w* 

21 12 1

 

each structure. The single-pass gain of the structure is given by [11]

other reasons, discussed below, four REs system is more appropriate.

, relax at a rate 2 (1 / ) *<sup>w</sup>*

, see Fig. 2. The Pauli blockings (1 ) *h*

, thus it does not contribute to gain. The escape

(6)

 

exp max int *g g aL <sup>s</sup>* (10)

*g*

(8)

(9)

(7)

*p*

max

. Both GS and ES are

to the

neglected. In WL, carriers are recombining at a rate (1 / ) *wR*

ES. ES relaxes quickly to GS at rate 21 (1 / )

assumed to be recombining at the same rate 1 (1 / ) *<sup>R</sup>*

component from ES to WL is 2 (1 / ) *<sup>w</sup>*

GS, respectively

*Sav* is given by [14]

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 rate (1 / ) *sr* , relax at rate (1 / )*<sup>s</sup>* to the WL which works as a common reservoir for the carriers. The escape rate from WL to the SCH is (1 / ) *we* . Other processes are described in 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* which are covered also in the three REs model

$$\frac{\partial N\_{\rm S}}{\partial t} = \frac{J}{qL\_{\rm w}} - \frac{N\_{\rm S}}{\tau\_{\rm s}} - \frac{N\_{\rm S}}{\tau\_{\rm sr}} + \frac{N\_{\rm w}}{\tau\_{\rm uve}} \tag{11}$$

$$\frac{\partial N\_w}{\partial t} = \frac{N\_S}{\tau\_s} - \frac{N\_w \{1 - h\}}{\tau\_w \tau\_2} + \frac{N\_w h}{\tau\_{2\,w}} - \frac{N\_w}{\tau\_{w\,R}} - \frac{N\_{w\,v}}{\tau\_{w\,e}} \tag{12}$$

$$\frac{\partial h}{\partial t} = \frac{N\_{\text{UV}}L\_{\text{UV}}\{1 - h\}}{N\_{\text{Q}}\tau\_{\text{W}2}} - \frac{N\_{\text{UV}}L\_{\text{UV}}h}{N\_{\text{Q}}\tau\_{\text{2}w}} - \frac{(1 - f)h}{\tau\_{\text{21}}} + \frac{f(1 - h)}{\tau\_{\text{12}}} \tag{13}$$

$$\frac{\partial f}{\partial t} = \frac{(1-f)h}{\tau\_{21}} - \frac{f(1-h)}{\tau\_{12}} - \frac{f^2}{\tau\_{1R}} - S\_{av} \Gamma \frac{g\_{\text{max}}\left(\hbar w\_p\right)}{N\_Q} (2f-1)L \frac{c}{n\_\mathcal{g}} \tag{14}$$

The Composition Effect on the Dynamics of Electrons in Sb-Based QD-SOAs 161

**1 InAsxSb1-x/GaAs0.1Sb0.9/Al0.1Ga0.9As 4 GaSb/GaAs0.7Sb0.3/GaAs** 

**2 In0.1AsSb0.9/GaAsxSb1-x/Al.1Ga.9As 5 InSb/GaAs0.7Sb0.3/GaAs** 

**Parameter Symbol (Unit) InSb GaSb InAs GaAs** 

( ) 0.17 0.73 0.36 1.424 **Bandgap energy** *E eV <sup>g</sup>*

0.44 0.33 0.41 0.45 *m m hh* / <sup>0</sup> **Heavy-hole Effective mass** 

**Refractive index** *n* 4 3.82 3.52 3.65

<sup>1</sup> ( ) 0.18 0.41 0.58 *E InAs Sb g xx x x*

<sup>1</sup> ( ) 0.726 0.502 1.2 *E GaAs Sb g xx x x*

Table 3. The relation used to calculate structure parameters (bandgap *Eg* and effective mass *mi*). Note that the subscript (i) with *mi* refers to conduction or valence band effective masses

( ) ( ) () *m ABC m AC m BC i ii*

( ) ( ) () *m ABC m AC m BC i ii*

<sup>1</sup> ( ) 1.242 1.247 *E Al Ga As gx x x* ( ) ( ) () *m ABC m AC m BC i ii*

0.0145 0.044 0.022 0.065 *m m <sup>e</sup>* / <sup>0</sup>

2

2

**No. Structure No. Structure** 

**3 In0.1AsSb0.9/GaAs0.1.Sb0.9/AlxGa1-xAs** 

Table 1. QD-SOA structures studied.

**Electron Effective Mass**

Table 2. Binary structure parameters [22].

**Structure Relation**

*x x* <sup>1</sup> *InAs Sb*

*GaAs Sb x x* <sup>1</sup>

*A x x* <sup>1</sup> *l Ga As*

[22, 23].

Quasi-thermal equilibrium is assumed between states. To ensure this convergence, the carrier escape times are related to the carrier capture times as follows [17]

$$
\pi\_{12} = \pi\_{d0}(\mu\_{\text{GS}} \mid \mu\_{\text{ES}})e^{\{[E\_{\text{ES}} - E\_{\text{CS}}]/k\_{\text{B}}T\}} \tag{15}
$$

$$
\pi\_{2w} = \pi\_{c0} (\mu\_{\rm ES} N\_Q \ne \rho\_{\rm weff}) e^{\langle \left[ E\_{\rm wf} - E\_{\rm ES} \right] / k\_B T \}} \tag{16}
$$

$$
\pi\_{\rm use} = \pi\_s(\rho\_{\rm new} N\_{\rm QD} \; / \; \rho\_{\rm SCH} H\_b) e^{\left(\Delta E\_{\rm SCH,ult} / k\_B T\right)} \tag{17}
$$

GS and ES QD energy levels are denoted by , *E E GS ES* . *weff* is the density of states per unit area in the WL and *SCH* is the density of states per unit volume in the SCH. They are given by <sup>2</sup> ( /) *weff ewl B m kT* and 2 3/2 2(2 / ) *SCH eSCH B m kT* . *NQD* is the number of QD layers and *Hb* is the total thickness of the SCH. *SCH wl* , *<sup>E</sup>* is the energy difference between SCH and WL band edge energies. The capture times *c*<sup>0</sup> and *<sup>d</sup>*<sup>0</sup> are the average capture time from the WL to the ES and from the ES to the GS with the hypothesis that the nal state is
