**1. Introduction**

Four-wave mixing (FWM) in semiconductor optical amplifiers (SOAs) has several important features, such as, high speed and high FWM conversion efficiency as well as optical demultiplexing (DEMUX) (Mecozzi *et al.,* 1995; Mecozzi & Mrk, 1997; Das *et al.,* 2000). The are several applications of FWM in SOAs for all-optical devices, such as, wavelength converters (Vahala *et al.,* 1996), optical samplers (Inoue & Kawaguchi*,* 1998), optical phase conjugators (Kikuchi & Matsumura*,* 1998) and optical multiplexers/demultiplexers (Kawanishi *et al.,* 1997; Kawanishi *et al.,* 1994; Morioka *et al.,* 1996; Uchiyama *et al.,* 1998; Tomkos *et al.,* 1999; Kirita *et al.,* 1998; Buxens *et al*., 2000) have been demonstrated for optical communication systems. When a pulse of a time-multiplexed signal train (for example, a probe pulse) and a pump pulse are injected simultaneously into an SOA, gain and refractive index in the SOA are modulated and an FWM signal is generated by the modulations. Thus, we can obtain a demultiplexed signal as an FWM signal at the output of SOA. All-optical demultiplexing has been experimentally demonstrated up to 200 Gbit/s (Morioka *et al.,* 1996). Tomkos *et al.,* (Tomkos *et al.,* 1999) suggested a number of ways to improve the performance of the dual-pump demultiplexer at 40 Gbit/s as follows; adjustment of the input wavelengths at the peak gain wavelength of the SOA under saturation conditions, the use of higher pump power at the input of the device, or/and the use of pulsed pumps with short pulsewidths. For the higher bit-rate, the overlap of the input to the FWM signal pulses may appear both in the time and spectral domain. The pattern effect may also appear in the FWM signal due to the slow components of the optical nonlinearities in SOAs (Saleh & Habbab, 1990). These effects degrade the usefulness of the FWM in SOAs as a practical DEMUX device in optical network/ communication systems. Therefore, it is very important to analysis the optical DEMUX characteristics based on FWM in SOAs for the ultrafast multi-bit input optical pulses.

The analyses of FWM in SOAs between short optical pulses have been widely reported (Shtaif & Eisenstein, 1995; Shtaif *et al.,* 1995; Xie *et al.,* 1999; Tang & Shore, 1998; Tang & Shore, 1999a; Tang & Shore, 1999b; Das *et al.,* 2000). The FWM conversion efficiency (Shtaif *et al.,* 1995; Xie *et al.,* 1999; Tang & Shore, 1998; Tang & Shore, 1999a; Mrk & Mecozzi, 1997)

Optical Demultiplexing Based on Four-Wave Mixing in Semiconductor Optical Amplifiers 167

Energy

= 0 < 100 fs 1 ps 1 ns

Fig. 1. Important nonlinear effects in SOAs are: (i) spectral hole-burning (SHB) with a life time of < 100 fs, (ii) carrier heating (CH) with a life time of ~ 1 ps, (iii) carrier depletion (CD)

Figure 1 shows the time-development of the population density in the conduction band after excitation (Das, 2000). The arrow (pump) shown in Fig. 1 is the excitation laser energy. Below the life-time of 100 fs, the SHB effect is dominant. SHB occurs when a narrow-band strong pump beam excites the SOA, which has an inhomogeneous broadening. SHB arises due to the finite value of intraband carrier-carrier scattering time (~ 50 – 100 fs), which sets the time scale on which a quasi-equilibrium Fermi distribution is established among the carriers in a band. After ~1 ps, the SHB effect is relaxed and the CH effect becomes dominant. The process tends to increase the temperature of the carriers beyond the lattice's temperature. The main causes of heating the carriers are (1) the stimulated emission, since it involves the removal of "cold" carriers close to the band edge and (2) the free-carrier absorption, which transfers carriers to high energies within the bands. The "hot"-carriers relax to the lattice temperature through the emission of optical phonons with a relaxation time of ~ 0.5 – 1 ps. The effect of CD remains for about 1 ns. The stimulated electron-hole recombination depletes the carriers, thus reducing the optical gain. The band-to-band relaxation also causes CD, with a relaxation time of ~ 0.2 – 1 ns. For ultrashort optical pumping, the two-photon absorption (TPA) effect also becomes important. An atom makes a transition from its ground state to the excited state by the simultaneous absorption of two laser photons. All these nonlinear effects (mechanisms) are taken into account in the simulation and the mathematical formulation of modified nonlinear Schrödinger equation

**2.2 Mathematical formulation of modified nonlinear schrödinger equation (MNLSE)**  In this subsection, we will briefly explain the theoretical analysis of short optical pulses propagation in SOAs. We start from Maxwell's equations (Agrawal, 1989; Yariv, 1991; Sauter, 1996) and reach the propagation equation of short optical pulses in SOAs, which are

governed by the wave equation (Agrawal & Olsson, 1989) in the frequency domain:

<sup>2</sup> (,,, ) (,,, ) 0 *<sup>r</sup> Exyz Exyz c* 

 

 

(1)

2 2

 

Population

Carrier Heating

Spectral Hole-Burning

with a life time is ~ 1 ns and (iv) two-photon absorption (TPA).

Energy

Energy

(MNLSE).

Population Population

Pump

Population

Carrier Depletion

Energy

the chirp of mixing pulses (Tang & Shore, 1999b; Das *et al.,* 2000), and the pump-probe time delay dependency of the FWM conversion efficiency (Shtaif & Eisenstein, 1995; Shtaif *et al.,*  1995; Mecozzi & Mrk, 1997; Das *et al.,* 2007; Das *et al.,* 2011) have been reported. On the contrary, however, there are only a few reports on analyses of FWM in SOAs used for demultiplexing time-division multiplexed data streams at ultra-high bit rates. Eiselt (Eiselt, 1995) reported the optimum control pulse energy and width with respect to the switching efficiency, channel crosstalk, and jitter tolerance. In those calculations, a very simple model of time-resolved gain saturation was used, which only took into account the gain recovery time. The FWM model was also very simple, in which the optical output power of the converted signal was proportional to the product of the squared pump output power and signal output power. Shtaif and Eisenstein (Shtaif & Eisenstein, 1996) calculated the error probabilities for time-domain DEMUX. Therefore, a detail and accurate analysis is required in order to clarify the performance of optical DEMUX based on FWM in SOAs for highspeed optical communication systems.

In this Chapter, we present detail numerical modeling/simulation results of FWM characteristics for the solitary probe pulse and optical DEMUX characteristics for multi-bit (multi-probe and/or pump) pulses in SOAs by using the finite-difference beam propagation method (FD-BPM) (Das *et al.,* 2000; Razaghi *et al.,* 2009). These simulations are based on the nonlinear propagation equation considering the group velocity dispersion, self-phase modulation (SPM), and two-photon absorption (TPA), with the dependencies on the carrier depletion (CD), carrier heating (CH), spectral-hole burning (SHB), and their dispersions, including the recovery times in SOAs (Hong *et al.,* 1996). For the simulation of solitary probe pulse, we obtain an optimum input pump pulsewidth from a viewpoint of ON/OFF ratios. For the simulation of optical DEMUX characteristics, we evaluate the ON/OFF ratios and the pattern effect of FWM signals for the multi-probe pulses. We have also simulated the optical DEMUX characteristics for the time-multiplexed signals by the repetitive pump pulses.

The FD-BPM is useful to obtain the propagation characteristics of single pulse or milti-pulses using the modified nonlinear Schrödinger equation (MNLSE) (Hong *et al.,* 1996 & Das *et al.,* 2000), simply by changing only the combination of input optical pulses. These are: (1) single pulse propagation (Das *et al.,* 2008), (2) FWM characteristics using two input pulses (Das *et al.,* 2000), (3) optical DENUX using several input pulses (Das *et al.,* 2001), (4) optical phaseconjugation using two input pulses with chirp (Das *et al.,* 2001) and (5) optimum time-delayed FWM characteristics between the two input pump and probe pulses (Das *et al.,* 2007).
