**2. Analytical model**

In this section, we briefly discuss the important nonlinear effects in SOAs, mathematical formulation of modified nonlinear Schrödinger equation (MNLSE), finite-difference beam propagation method (FD-BPM) that is used in the simulation, and nonlinear propagation characteristics of solitary optical pulses in SOAs.

#### **2.1. Important nonlinear effects in SOAs**

86 Optical Communication

Kawaguchi & Inoue, 1996b).

(Das *et al.,* 2007).

**2. Analytical model** 

although they may include waveform distortion and additional chirp. If the demultiplexed signal obtained as the FWM signal has optical phase-conjugate characteristics, the demultiplexed signal can be compressed using a dispersive medium. However, such optical phase-conjugate characteristics of FWM signals have not yet been reported to the best of author's knowledge. Another advantage of FWM with short duration optical pump pulses is the high conversion efficiency. In conventional systems, the FWM conversion efficiency in an SOA is limited due to gain saturation. However, high FWM conversion efficiency can be achieved with short duration optical pump and probe pulses as it is possible to reduce gain saturation and hence increase FWM conversion efficiency in an SOA by applying strong pump intensity (Shtaif & Eisenstein, 1995; Shtaif *et al.,* 1995; Kawaguchi & Inoue, 1996a;

In this chapter, we present the detail numerical simulation results of optical phase-conjugate characteristics of picosecond FWM signal pulses generated in SOAs using the FD-BMP (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 twophoton 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). The main purpose of our simulations is to provide answers to the following questions: 1) how is the nature of the optical phase-conjugate maintained for a short FWM signal pulse? 2) how does the chirp observed in the FWM signals affect the nature of the optical phase-conjugate? For this reason, we have analyzed the system in which the Fourier transform-limited Gaussian optical pulse is linearly chirped by transmission through a fiber (Fiber I) and then injected into an SOA as a probe pulse, together with a pump pulse that has a 1 ~ 10 ps pulsewidth. The FWM signal is generated by the mixing of the pump pulse and the probe pulse, and is selected by an optical narrow band-pass filter. The FWM signal is then transmitted through another fiber (Fiber II) that has the same group velocity dispersion (GVD) as Fiber I and an appropriate length. The simulations are based on the nonlinear propagation equation considering the GVD, SPM, and TPA, with dependencies on CD, CH, SHB, and

The FD-BPM is useful to obtain the propagation characteristics of single pulse or miltipulses 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 phase-conjugation 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

In this section, we briefly discuss the important nonlinear effects in SOAs, mathematical formulation of modified nonlinear Schrödinger equation (MNLSE), finite-difference beam

dispersion of those properties (Das *et al.,* 2000; Hong *et al.,* 1996).

There are several types of "nonlinear effects" occurs in SOAs. Among them, the important four types of "nonlinear effects" are shown in Fig. 1. These are namely: (i) spectral holeburning (SHB), (ii) carrier heating (CH), (iii) carrier depletion (CD) and (iv) two-photon absorption (TPA).

**Figure 1.** The important nonlinear effects in SOAs: (i) spectral hole-burning (SHB) with a life time of less than 100 fs (i.e., < 100 fs), (ii) carrier heating (CH) with a life time of ~ 1 ps, (iii) carrier depletion (CD) with a life time is ~ 1 ns and (iv) two-photon absorption (TPA).

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. When the life time is less than 100 fs (i.e., < 100 fs), the SHB effect is dominant. The SHB occurs when a narrow-band strong pump beam excites the SOA, which has an inhomogeneous broadening. The SHB arises due to the finite value of intraband carriercarrier 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 the life time ~ 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 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

modeling/ simulation and the mathematical formulation of modified nonlinear Schrödinger equation (MNLSE).
