**1. Introduction**

The application of four-wave mixing (FWM) in semiconductor optical amplifiers (SOAs) has been widely demonstrated to all-optical devices, such as wavelength converters (Vahala *et al.,* 1996; Nesset *et al.,* 1998), optical samplers (Kawaguchi & Inoue, 1998), optical multiplexers/ demultiplexers (Kawanishi *et al.,* 1997; Uchiyama *et al.,* 1998; Tomkos *et al.,*  1999; Buxens *et al*., 2000; Set *et al*., 1998), and optical phase conjugators (Dijaili *et al.,* 1990; Kikuchi & Matsumura*,* 1998; Marcenac *et al.,* 1998; Corchia *et al.,* 1999; Tatham *et al.,* 1993; Ducellier *et al.,* 1996), which are expected to be used in future optical communication systems. Kikuchi & Matsumura have demonstrated the transmission of 2-ps optical pulses at 1.55 μm over 40 km of standard fiber by employing midspan optical phase conjugation in SOAs (Kikuchi & Matsumura*,* 1998). An ideal phase conjugator must reverse the chirp of optical pulses while maintaining the pulse waveform. Kikuchi & Matsumura (Kikuchi & Matsumura*,* 1998) have shown that the second-order dispersion is entirely compensated by the optical phase-conjugation obtained using SOAs with a continuous wave (cw) input pump wave. All-optical demultiplexing (DEMUX), based on FWM in SOAs, was also demonstrated. When a single pulse of a time-multiplexed signal train (as probe pulses) and a pump pulse are injected simultaneously into an SOA, the gain and refractive index in the SOA are modulated, and an FWM signal pulse is created by the modulations. Thus, we can obtain a demultiplexed signal as the FWM signal by Das *et al.,* (Das *et al.,* 2000). All-optical DEMUX has been experimentally demonstrated up to 200 Gbit/s by Kawanishi *et al.,* (Kawanishi *et al.,* 1997). Many research reports have been published recently on the theoretical investigaton of the characteristics of FWM for short optical pulses in SOAs. Tang and Shore (Tang & Shore, 1999) theoretically examined the dynamical chirping of mixing pulses and showed that all mixing pulses have negative pulse chirp except in the far edges of trailing pulses, indicating that pulse spectra are primarily red-shifted. The demultiplexed signals obtained as FWM signals may still have optical phase-conjugate characteristics,

© 2012 Das et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Das et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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; Kawaguchi & Inoue, 1996b).

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

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 (Das *et al.,* 2007).
