**2.2 Numerical method**

In order to solve the model numerically, we divide the SOA into *Nz* sections of equal length in the optical active waveguide, thus having a section length of *z* = *L*/*Nz*, and choose a corresponding time resolution of *t* = *z*/*ng*. *Nz* should be large enough to have a good numerical approximation.


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

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).

High-Speed All-Optical Switches Based on Cascaded SOAs 31

So far, we have presented a detailed model of a single SOA. The simulation of the cascaded-SOA-switch can be completed by the following calculation procedure, where we use the

1. Calculate the steady state of SOA1, and obtain the initial state of the carrier densities, ASE in each section, which will be used as initial conditions in following calculations. In the case of the steady state, the RHS of (1)-(3) should equal to zero, which implies that the carrier densities in each section will keep unchanging if the input does not change. A numerical algorithm from (Connelly, 2001) is adopted here to give a good

2. Calculate the response of SOA1 and get the output, by applying a proper input optical signal like a pump pulse or a pseudo-random binary sequence (PRBS) modulated pump pulse train, in addition to the probe CW beam. Firstly, according to initial conditions at time step *t*1 obtained from step 1), carrier densities *N*(*zi*, *t2*) can be calculated by using (11), so does *nCH*(*zi*, *t2*), *nSHB* (*zi*, *t2*), and the optical signal and ASE power in each section at the time step *t2* from (12-15). Thereby, all necessary quantities of SOA1 at time step *t2* are obtained, which can be treated as initial conditions to further calculations of the next time step. As the iteration completes, the *N* and *P* at each section and each time step can

3. Filter out the pump pulse or PRBS, and only allow the modulated CW signal to enter

4. Repeat step 1) to get the initial steady conditions of SOA2 firstly. It should be mentioned that, under this circumstance the amplified CW after SOA1 has to be used as the input to SOA2 to obtain the initial carrier densities and ASE levels in each section of

5. Repeat step 2) using the modulated CW signal from the output of SOA1, as the input to SOA2. Calculate the output of SOA2, which is the final output of the turbo-switch.

The parameters used in our model are list in Table I. Two identical SOAs are applied in all the turbo-switch simulation, as implemented in the reported experiments. The SOAs are 0.7 mm long, which have a relatively high gain and short carrier lifetime, as well as an acceptable noise figure. A 200 mA bias current is consistently used unless specifically described. A 100% of the injected current utilization is supposed in the model. In the following simulations, the input CW and pump pulse are at wavelengths of 1560 and 1550

A steady state numerical algorithm presented in (Connelly, 2001) is used to obtain the SOA gain saturation characteristics, as illustrated in Fig. 3, where the wavelength of the CW input beam is 1560 nm. It is shown that, the small-signal gain of the amplifier is 25 dB, while the

nm respectively, and the pulsewidth is 3 ps (FWHM) if not otherwise specified.

be obtained, which gives the output of the SOA1, *P*(*zNz+1*, *tj*).

where the facet reflection is neglected.

case of 2 SOAs (turbo-switch) as the example:

**2.3 Simulation procedure** 

convergence.

the SOA2.

SOA2.

**3. Simulation results** 

saturation output power is 12 dBm.


Optical powers and ASE propagating in the positive and negative directions are calculated at the boundaries of each section, while the total carrier density and local carrier changes caused by the CH and SHB processes are considered at the center of each section. When the time interval *t* is small enough, the left hand side (LHS) of (1) can be approximated by,

$$\frac{\partial \mathcal{N}(z\_i, t\_j)}{\partial t} = \frac{\mathcal{N}(z\_i, t\_j) - \mathcal{N}(z\_i, t\_{j-1})}{\Delta t} \tag{10}$$

Thus, basing upon the carrier density and the photon densities at the previous time step, we have,

$$N(z\_i, t\_j) = N(z\_i, t\_{j-1}) + \Delta t \left[ \frac{I}{eV} + R(N(z\_i, t\_{j-1})) - v\_\mathcal{g} \mathcal{g} \frac{S(z\_i, t\_{j-1}) + S(z\_{i+1}, t\_{j-1})}{2} \right.$$

$$-v\_\mathcal{g} \mathcal{g}\_{\text{ase}} \frac{S\_{\text{ase}}^+(z\_i, t\_{j-1}) + S\_{\text{ase}}^+(z\_{i+1}, t\_{j-1})}{2} - v\_\mathcal{g} \mathcal{g}\_{\text{ase}} \frac{S\_{\text{ase}}(z\_i, t\_{j-1}) + S\_{\text{ase}}^-(z\_{i+1}, t\_{j-1})}{2} \right] \tag{11}$$

where a linear interpolation is employed to estimate the photon densities of the input optical beam, co-propagating and counter-propagating ASEs at the center of each section. Similar method can be applied to (2) and (3), to calculate the local carrier density variations due to CH and SHB processes.

The first term in the LHS of (7), describes the optical power propagating along the *z*-axis of the SOA, and experiencing an exponential amplification by a factor of (*g* - *int*), as shown in the RHS, which can be assumed constant in a sufficiently small interval *z*. The second term in the LHS, however, accounts for the optical power variation during the travelling time period in the section, which can be included using values obtained at last time step (Bischoff, 2004). Therefore, a solution of (7) is,

$$P(z\_{i+1}, t\_j) = P(z\_i, t\_{j-1}) \exp\left| \left( F \lg \mathcal{N}(z\_i, t\_{j-1}) - a\_{\text{int}} \right) \Delta z \right| \tag{12}$$

subjected to boundary condition,

$$P(z\_1, t\_j) = P\_{in}(t\_j) \tag{13}$$

where *Pin*(*tj*) denotes the input optical power at *tj*.

Similar solutions can be given for the co-propagating and the counter-propagating ASEs, as described in (8),

$$P\_{\rm as\epsilon}^{+}(\mathbf{z}\_{i+1},t\_{j}) = P\_{\rm as\epsilon}^{+}(\mathbf{z}\_{i},t\_{j-1}) \exp\left\{g\_{\rm as\epsilon}^{\prime}\Delta\mathbf{z}\right\} + \left[\beta\_{\rm as\epsilon}R\_{sp}(\mathcal{N}(\mathbf{z}\_{i},t\_{j-1}))\frac{hc}{\lambda\_{\rm as\epsilon}}\frac{\delta}{\Gamma}\right] \frac{\exp\left\{g\_{\rm as\epsilon}^{\prime}\Delta\mathbf{z}\right\} - 1}{g\_{\rm as\epsilon}^{\prime}} \tag{14a}$$

$$P\_{\rm ase}^{-}(\mathbf{z}\_{i},t\_{j}) = P\_{\rm ase}^{-}(\mathbf{z}\_{i+1},t\_{j-1}) \exp\left\{g\_{\rm ase}^{\prime}\Delta\mathbf{z}\right\} + \left[\mathcal{J}\_{\rm ase}R\_{sp}(\mathcal{N}(\mathbf{z}\_{i},t\_{j-1}))\frac{hc}{\lambda\_{\rm ase}}\frac{\delta}{\Gamma}\right] \frac{\exp\left\{g\_{\rm ase}^{\prime}\Delta\mathbf{z}\right\} - 1}{g\_{\rm ase}^{\prime}} \tag{14b}$$

where 1 int ( ( , ), ) *g gNz t ase i j ase* , and subjected to boundary conditions respectively,

$$P\_{ase}^{+}(z\_1, t\_j) = 0\tag{15a}$$

$$P\_{\rm ase}^{\cdot}(z\_{\rm Nz+1}, t\_j) = 0 \tag{15b}$$

where the facet reflection is neglected.
