**2.1 SOA model**

The basic time-domain rate equations describing the carrier dynamics via the inter-band and intra-band processes in a single SOA, as proposed in (Gutiérrez-Castrejón, 2009; Mecozzi & 26 Selected Topics on Optical Amplifiers in Present Scenario

85Gb/s, where dual ultrafast nonlinear interferometers (UNIs) were implemented (Yang et

Fig. 1. Schematic setup of the turbo-switch, where the OBF is used to remove the pump

In this chapter, we will review the recent progress of the all-optical high-speed switches using cascaded SOAs, from both theoretical and experimental aspects. A majority of the publications (Manning et al., 2006, 2007; Yang et al., 2006, 2010) related to turbo-switch were reported, showing the high-speed experimental performances of turbo-switch over a single SOA. Apparently, a systematic theoretical turbo-switch model is necessary for the purpose of understanding the fundamental behaviors of the turbo-switch and how to further enhance the switch performance. First of all, we will present a detailed time-domain SOA model, from which the turbo-switches and switches with three or more cascaded SOAs can be evaluated. For the reason of convenience, we will refer hereafter to this kind of switch, including turbo-switch, as cascaded-SOA-switch. Then, we will focus on the relation between the overall performance of the switch and the nonlinear gain/refractive-index dynamics of the individual SOAs. The amplitude/phase dynamics of the optical output signal from the switch will be analyzed in details and compared with the experimental data. The SOA model will certainly help us not only to understand the basic principles of the switch, but also to exploit the way and the critical conditions for the switch to operate at

The chapter is organized as follows. Section 2 presents a comprehensive theoretical analysis of the cascaded-SOA-switch, where the SOA model and the corresponding simulation method are presented. Simulation results including the gain/phase dynamics, pattern effect mitigation using turbo-switch, are shown in Section 3. Experimental demonstrations of 170 Gb/s AND gate (wavelength conversion) and 85 Gb/s XOR gate using turbo-switches are presented in Section 4. The cascaded-SOA-switches are further exploited in terms of the number of cascaded SOAs in Section 5, where the overall gain recovery time, the noise figure as well as the impact of injected SOA current of the cascaded switches are illustrated

in details, as simulated by the model. Finally, conclusions will be given in Section 6.

To explore the operation principle and understand the performances of the cascaded-SOAswitch, a time-domain SOA model is required to analyze the fundamental gain/phase behaviors of the SOA-based device as well as to simulate the speed and application of the

The basic time-domain rate equations describing the carrier dynamics via the inter-band and intra-band processes in a single SOA, as proposed in (Gutiérrez-Castrejón, 2009; Mecozzi &

al., 2006, 2010) and the turbo-switch configuration was incorporated.

signal. OBF: optical band-pass filter.

even higher bit-rates.

devices.

**2.1 SOA model** 

**2. Theoretical analysis of SOAs** 

Mørk, 1997), are adopted. Travelling-wave equations in terms of the optical amplitude/power and phase, derived from Maxwell equations and Kramers-Kronig relations, are also incorporated in the SOA model to obtain the amplitude and phase of the output optical signal propagating through the SOA (Mecozzi & Mørk, 1997; Agrawal & Olsson, 1989).

Following the SOA model in (Mecozzi & Mørk, 1997), rate equations for the total carrier density *N* related to the (inter-band) band-filling effect, and the local carrier density variations *nCH* and *nSHB*, which are associated with the ultrafast (intra-band) effects: carrier heating (CH) and spectrum hole burning (SHB) processes respectively, can be expressed as follows:

$$\frac{\partial \mathcal{N}(z,t)}{\partial t} = \frac{I}{eV} - \mathcal{R}(\mathcal{N}(z,t)) - v\_g \mathcal{g} \mathcal{S}(z,t) - v\_g \mathcal{g}\_{\text{asec}} \left[ \mathcal{S}^+\_{\text{asec}}(z,t) + \mathcal{S}^-\_{\text{asec}}(z,t) \right] \tag{1}$$

$$\frac{\partial n\_{\rm CH}(z,t)}{\partial t} = -\frac{n\_{\rm CH}(z,t)}{\tau\_{\rm CH}} - \frac{\kappa\_{\rm CH}}{a\_0 \tau\_{\rm CH}} \text{g} \, \text{S}(z,t) \tag{2}$$

$$\frac{\partial n\_{\rm SHB}(z,t)}{\partial t} = -\frac{n\_{\rm SHB}(z,t)}{\tau\_{\rm SHB}} - \frac{\varepsilon\_{\rm SHB}}{a\_0 \tau\_{\rm SHB}} \, \text{g} \, \text{S}(z,t) - \left[\frac{\partial \text{N}(z,t)}{\partial t} + \frac{\partial n\_{\rm CH}(z,t)}{\partial t}\right] \tag{3}$$

where the first term in the right hand side (RHS) of (1) represents the increase of the total carrier density due to the injected current *I* to the SOA. Here, we have assumed a uniform distribution of the injected current along the longitude. In (1), *e* is the electron charge, and *V* is the volume of the active region in the SOA.

The radiative and nonradiative recombination rate due to the limited carrier lifetime in the SOA, *R*(*N*) (Connelly, 2001), can be approached by,

$$R(\text{N}) = A\text{N} + B\text{N}^2 + \text{CN}^3 \tag{4}$$

where *A*, *B*, *C* represent the linear, bimolecular, and auger recombination coefficients respectively.

The third and fourth terms in the RHS of (1) are used to account for the depletion of total carrier density aroused from the stimulation emission by the injected light and the amplified spontaneous emission (ASE), respectively. *vg* is the group velocity. *g* is the gain coefficient and *S* is the photon density in the active region. *gase* is the equivalent gain coefficient for ASE (Talli & Adams, 2003). *CH* and *CH* in (2) are carrier-carrier relaxation time and gain suppression factor caused by CH, while *SHB* and *SHB* in (3) are temperature relaxation time and gain suppression factor caused by SHB.

To take the gain dispersion into account better, and make our model applicable in a wide optical wavelength range, a polynomial model for the gain coefficient (Leuthold et al., 2000), which combines of a quadratic and a cubic function, is used, with one modification to include the ultrafast effect induced by CH and SHB.

$$\mathbf{g} = \begin{cases} \mathbf{g}\_l + \mathbf{g}\_{h'} & \mathcal{k} < \mathcal{k}\_{\mathbb{Z}}(\mathcal{N}) \\ \mathbf{0}\_{\prime} & \mathcal{k} \ge \mathcal{k}\_{\mathbb{Z}}(\mathcal{N}) \end{cases} \tag{5a}$$

$$\mathcal{S}\_{\beta} = \mathcal{c}\_{N,\beta} \left[ \mathcal{J} - \mathcal{J}\_{\mathbf{z}}(N) \right]^2 + d\_{N,\beta} \left[ \mathcal{J} - \mathcal{J}\_{\mathbf{z}}(N) \right]^3 \tag{5b}$$

High-Speed All-Optical Switches Based on Cascaded SOAs 29

where an additional term in the RHS, comparing to (7), is used to account for the spontaneous emission (SE) coupled into the effective waveguide. *Rsp* = *BN2* is the SE rate. "+" stands for the co-propagating direction with the input light, while "-" represents the

Carrier density variations not only affect the gain, but also change the phase of the input optical signal. Associated with the gain dynamics through Kramers-Kronig relations, the phase shift (Mecozzi & Mørk, 1997) of the optical beam due to the SOA nonlinearity can be

2

filling and CH process, respectively. The subscript *nSHB* = 0 means the SHB impact on the

It should be mentioned that, many physical effects of the SOA, including two-photon absorption (TPA), ultrafast nonlinear refraction (UNR), free-carrier absorption (FCA) and group velocity dispersion (GVD), are neglected in our SOA model. Ultrafast processes such as TPA, FCA and UNR are ignored reasonably, because these effects become important only when pulse energy is stronger than 1 pJ (Yang et al., 2003), while the pulse energy used in our simulation is generally lower than 0.1 pJ. GVD is also neglected, since the Gaussian pump pulsewidth (full width at half maximum, FWHM) in the paper is assumed to be 2~3 ps, which means that the spectral detuning from the central frequency is less than a few THz

In order to solve the model numerically, we divide the SOA into *Nz* sections of equal length

( 1) *<sup>i</sup> zi z <sup>i</sup>* <sup>1</sup> *z iz*

Fig. 2 shows a sketch of the *i*th section in the SOA, where *i*=1,2,…, *Nz* and *j*=1,2,…,*Nt*. *Nz* and *Nt* are the total number of the SOA sections and time steps respectively (Connelly, 2001).

(,) (,) (,)

*Nzt n zt n zt*

*i j CH i j SHB i j*

<sup>1</sup> ( ,) *ase <sup>i</sup> <sup>j</sup> <sup>P</sup> z t* (,) *ase i j P zt*

*N l T hnSHB*

*ase ase ase ase ase sp*

(,) (,) <sup>1</sup> ( ) (,)

(,) 1 (,) 1

 

*g zt zt*

in the optical active waveguide, thus having a section length of

(,) (,) *i j ase i j*

*Pzt P zt*

Fig. 2. A schematic sketch of the *i*th section of the SOA.

*t* = 

counter-propagating direction.

phase shift is ignored here.

(Mecozzi & Mørk, 1997).

**2.2 Numerical method** 

numerical approximation.

corresponding time resolution of

expressed as,

where *N* and *T* is the  int

*g ase P zt P zt hc g P zt R zv t* (8)

 

, 0

*z*/*ng*. *Nz* should be large enough to have a good

1 1

( ,) ( ,) *i j ase i j*

*Pz t P zt* 

*z* = *L*/*Nz*, and choose a

 


*g g zv t* (9)

where  *= l, h* represents the gain coefficient attributed to total carrier density *N* and CH/SHB effect, respectively.

Polynomial coefficients are calculated by,

$$\mathcal{L}\_{N,\beta} = 3 \frac{\mathcal{S}\_{p,\beta}}{\left[\mathcal{A}\_{\mathbb{Z}}(N) - \mathcal{A}\_{p}(N)\right]^2} \tag{5c}$$

$$d\_{N, \beta} = 2 \frac{\mathcal{S}\_{p, \beta}}{\left[\mathcal{A}\_z(N) - \mathcal{A}\_p(N)\right]^3} \tag{5d}$$

where *gp*, , *p*(*N*) and *<sup>z</sup>*(*<sup>N</sup>*) stand for the material gain at the peak wavelength, the shifted wavelength at peak and transparency respectively. They are approximated by,

$$\mathcal{g}\_{p,l} = a\_0(N - N\_0) + \overline{a}a\_0N\_0e^{-\mathcal{N}\_{\mathbb{K}\_0}} \tag{5e}$$

$$\mathbf{g}\_{p,h} = \mathbf{a}\_0 (\mathbf{n}\_{\rm CH} + \mathbf{n}\_{\rm SHB}) \tag{5f}$$

$$\mathcal{A}\_p(\text{N}) = \mathcal{A}\_{p\_0} - \left[ b\_0(\text{N} - \text{N}\_0) + b\_1(\text{N} - \text{N}\_0)^2 \right] \tag{5g}$$

$$\mathcal{A}\_{\mathbf{z}}(\mathbf{N}) = \mathcal{A}\_{\mathbf{z}\_0} - z\_0 (\mathbf{N} - \mathbf{N}\_0) \tag{5h}$$

where *a0*, *N0*, *a* , *p0*, *b0*, *b1*, *z0*, and *z0* are parameters which have to be obtained by experimental gain dispersion curves (Leuthold et al., 2000). *N0* represents the transparency carrier density at the peak wavelength *0*.

By definition, the photon density *S* (in unit of *m-3*) in (1)-(3) can be expressed in terms of the light power *P* (in unit of *W*) as,

$$S(z,t) = \frac{P(z,t)}{h(c/\lambda)(\delta / \varGamma)v\_{\mathcal{S}}} \tag{6}$$

where *h*, *c*, denotes for Planck's constant, speed of light in vacuum, cross section area of the active region and confinement factor, respectively.

The travelling-wave equation of the input optical light (Agrawal & Olsson, 1989) is,

$$\frac{\partial P(z,t)}{\partial z} + \frac{1}{v\_{\mathcal{g}}} \frac{\partial P(z,t)}{\partial t} = (\Gamma \, \mathcal{g} - \alpha\_{\text{int}}) P(z,t) \tag{7}$$

where the power *P* is a function of time *t* and position *z* along the active waveguide (*z*-axis) of the SOA. *int* is the internal loss in the active region. Eq. (7) only represents the positive direction propagation of the input light, since the facet reflection of the SOA (below 10-4) is usually ignorable (Dutta & Wang, 2006).

For the propagation of the ASE power inside the amplifier, a bi-directional model presented in (Talli & Adams, 2003) is adopted, where the ASE is described by its total power while neglecting its spectral dependency. Equivalent coupling efficiency *ase*, equivalent wavelength *ase*, and equivalent gain coefficient *gase*, are used in the calculation, for the reason of computational efficiency.

0

*0*.

 

*N*

wavelength at peak and transparency respectively. They are approximated by,

*<sup>g</sup> <sup>d</sup>*

*N*

*c*

 *= l, h* represents the gain coefficient attributed to total carrier density *N* and

, , <sup>2</sup> 3 () () 

(5c)

(5d)

 

, , <sup>3</sup> 2 () () 

 

0 01 0

<sup>0</sup> 0 0

*z z* () ( ) *N zN N* 

experimental gain dispersion curves (Leuthold et al., 2000). *N0* represents the transparency

By definition, the photon density *S* (in unit of *m-3*) in (1)-(3) can be expressed in terms of the

(,) (,) ( / )( / ) 

*Pzt Szt*

The travelling-wave equation of the input optical light (Agrawal & Olsson, 1989) is,

 

*<sup>g</sup>*

neglecting its spectral dependency. Equivalent coupling efficiency

(,) 1 (,) ( ) (,) 

where the power *P* is a function of time *t* and position *z* along the active waveguide (*z*-axis)

direction propagation of the input light, since the facet reflection of the SOA (below 10-4) is

For the propagation of the ASE power inside the amplifier, a bi-directional model presented in (Talli & Adams, 2003) is adopted, where the ASE is described by its total power while

*<sup>z</sup>*(*<sup>N</sup>*) stand for the material gain at the peak wavelength, the shifted

0

() ( ) ( ) *p p N bN N bN N* (5g)

*g*

int

*Pzt Pzt g Pzt zv t* (7)

denotes for Planck's constant, speed of light in vacuum, cross section area of

*int* is the internal loss in the active region. Eq. (7) only represents the positive

*ase*, and equivalent gain coefficient *gase*, are used in the calculation, for the

*p l g a N N aa N e* (5e)

, 0 *g an n <sup>p</sup> h CH SHB* ( ) (5f)

2

*z0*, and *z0* are parameters which have to be obtained by

(5h)

*hc v* (6)

*ase*, equivalent

*N N*

 *p*

*z p*

, 0 0 00 ( ) *<sup>N</sup> <sup>N</sup>*

*N N*

 *p*

*z p*

*g*

where

where *gp*,

, *p*(*N*) and

where *a0*, *N0*, *a* ,

where *h*, *c*,

of the SOA.

wavelength

CH/SHB effect, respectively.

Polynomial coefficients are calculated by,

*p0*, *b0*, *b1*,

the active region and confinement factor, respectively.

carrier density at the peak wavelength

light power *P* (in unit of *W*) as,

reason of computational efficiency.

usually ignorable (Dutta & Wang, 2006).

$$\frac{\partial P\_{\rm as\epsilon}^{\pm}(z,t)}{\partial z} \pm \frac{1}{v\_{\rm g}} \frac{\partial P\_{\rm as\epsilon}^{\pm}(z,t)}{\partial t} = \pm (\Gamma \, g\_{\rm as\epsilon} - a\_{\rm int}) P\_{\rm as\epsilon}^{\pm}(z,t) \pm \beta\_{\rm as\epsilon} R\_{\rm sp} \frac{hc}{\lambda\_{\rm use}} \frac{\delta}{\Gamma} \tag{8}$$

where an additional term in the RHS, comparing to (7), is used to account for the spontaneous emission (SE) coupled into the effective waveguide. *Rsp* = *BN2* is the SE rate. "+" stands for the co-propagating direction with the input light, while "-" represents the counter-propagating direction.

Carrier density variations not only affect the gain, but also change the phase of the input optical signal. Associated with the gain dynamics through Kramers-Kronig relations, the phase shift (Mecozzi & Mørk, 1997) of the optical beam due to the SOA nonlinearity can be expressed as,

$$\frac{\partial \phi(z,t)}{\partial z} + \frac{1}{v\_{\mathcal{g}}} \frac{\partial \phi(z,t)}{\partial t} = -\frac{1}{2} \Gamma \left[ a\_N \mathcal{g}\_l + a\_T \mathcal{g}\_{h, n\_{S \to \partial}} = 0 \right] \tag{9}$$

where *N* and *T* is the -factors (also known as linewidth enhancement factor) for the bandfilling and CH process, respectively. The subscript *nSHB* = 0 means the SHB impact on the phase shift is ignored here.

It should be mentioned that, many physical effects of the SOA, including two-photon absorption (TPA), ultrafast nonlinear refraction (UNR), free-carrier absorption (FCA) and group velocity dispersion (GVD), are neglected in our SOA model. Ultrafast processes such as TPA, FCA and UNR are ignored reasonably, because these effects become important only when pulse energy is stronger than 1 pJ (Yang et al., 2003), while the pulse energy used in our simulation is generally lower than 0.1 pJ. GVD is also neglected, since the Gaussian pump pulsewidth (full width at half maximum, FWHM) in the paper is assumed to be 2~3 ps, which means that the spectral detuning from the central frequency is less than a few THz (Mecozzi & Mørk, 1997).
