**2.4. Nonlinear pulse propagation model in SOAs**

Nonlinear optical pulse propagation in SOAs has drawn considerable attention due to its potential applications in optical communication systems, such as a wavelength converter based on FWM and switching. The advantages of using SOAs include the amplification of small (weak) optical pulses and the realization of high efficient FWM characteristics.

For the analysis of optical pulse propagation in SOAs using the FD-BPM in conjunction with the MNLSE, where several parameters are taken into account, namely, the group velocity dispersion, SPM, and TPA, as well as the dependencies on the CD, CH, SHB and their dispersions, including the recovery times in an SOA (Hong *et al.,* 1996). We also considered the gain spectrum (as shown in Fig. 2). The gain in an SOA was dynamically changed depending on values used for the carrier density and carrier temperature in the propagation equation (i.e., MNLSE).

Initially, (Hong *et al.,* 1996) used the MNLSE for the simulation of optical pulse propagation in an SOA by FFT-BPM (Okamoto, 1992; Brigham, 1988) but the dynamic gain terms were changing with time. The FD-BPM is capable to simulate the optical pulse propagation taking into consideration the dynamic gain terms in SOAs (Das *et al.,* 2000 & 2007; Razaghi *et al.,* 2009a & 2009b; Aghajanpour *et al.,* 2009). We used the modified MNLSE for nonlinear optical pulse propagation in SOAs by the FD-BPM (Chung & Dagli, 1990; Conte & Boor, 1980). We used the FD-BPM for the simulation of optical phase-conjugation characteristics of picosecond FWM signal pulses in SOAs.

**Figure 4.** A simple schematic diagram for the simulation of nonlinear single pulse propagation in SOA. Here, <sup>2</sup> *V*( ,0) τ is the input (z = 0) pulse intensity and <sup>2</sup> *V z* (,) τ is the output pulse intensity (after propagating a distance z) of SOA.

Figure 4 illustrates a simple model for the simulation of nonlinear optical pulse propagation in an SOA. An optical pulse is injected into the input side of the SOA (z = 0). Here, τ is the local time, <sup>2</sup> *<sup>V</sup>*( ,0) τ is the intensity (power) of input pulse (z = 0) and <sup>2</sup> *V z* (,) τ is the intensity (power) of the output pulse after propagating a distance **z** at the output side of SOA. We used this model to simulate the FWM and optical phase-conjugation characteristics in SOAs.

#### **3. FWM characteristics between optical pulses in SOAs**

94 Optical Communication

equation (i.e., MNLSE).

Here,

<sup>2</sup> *V*( ,0) τ

local time, <sup>2</sup> *<sup>V</sup>*( ,0)

propagating a distance z) of SOA.

τ

picosecond FWM signal pulses in SOAs.

.

Input Pulse

*|V(*τ*, 0)|<sup>2</sup>*

input pump and probe pulses (Das *et al.,* 2007).

**2.4. Nonlinear pulse propagation model in SOAs** 

*et al.,* 2001), and (5) FWM conversion efficiency with optimum time-delays between the

Nonlinear optical pulse propagation in SOAs has drawn considerable attention due to its potential applications in optical communication systems, such as a wavelength converter based on FWM and switching. The advantages of using SOAs include the amplification of

For the analysis of optical pulse propagation in SOAs using the FD-BPM in conjunction with the MNLSE, where several parameters are taken into account, namely, the group velocity dispersion, SPM, and TPA, as well as the dependencies on the CD, CH, SHB and their dispersions, including the recovery times in an SOA (Hong *et al.,* 1996). We also considered the gain spectrum (as shown in Fig. 2). The gain in an SOA was dynamically changed depending on values used for the carrier density and carrier temperature in the propagation

Initially, (Hong *et al.,* 1996) used the MNLSE for the simulation of optical pulse propagation in an SOA by FFT-BPM (Okamoto, 1992; Brigham, 1988) but the dynamic gain terms were changing with time. The FD-BPM is capable to simulate the optical pulse propagation taking into consideration the dynamic gain terms in SOAs (Das *et al.,* 2000 & 2007; Razaghi *et al.,* 2009a & 2009b; Aghajanpour *et al.,* 2009). We used the modified MNLSE for nonlinear optical pulse propagation in SOAs by the FD-BPM (Chung & Dagli, 1990; Conte & Boor, 1980). We used the FD-BPM for the simulation of optical phase-conjugation characteristics of

**Figure 4.** A simple schematic diagram for the simulation of nonlinear single pulse propagation in SOA.

SOA

Figure 4 illustrates a simple model for the simulation of nonlinear optical pulse propagation

intensity (power) of the output pulse after propagating a distance **z** at the output side of

in an SOA. An optical pulse is injected into the input side of the SOA (z = 0). Here,

<sup>2</sup> *V z* (,) τ

is the intensity (power) of input pulse (z = 0) and <sup>2</sup> *V z* (,)

is the output pulse intensity (after

*|V(*τ*, z)|<sup>2</sup>*

Output Pulse

. <sup>τ</sup>

0

τis the

is the

τ

is the input (z = 0) pulse intensity and

0 τ

small (weak) optical pulses and the realization of high efficient FWM characteristics.

In this section, we will discuss the FWM characteristics between optical pulses in SOAs. When two optical pulses with different central frequencies *fp* (pump) and *fq* (probe) are injected simultaneously into the SOA, an FWM signal is generated at the output of the SOA at a frequency of 2*fp* - *fq* (as shown in Fig. 5). For the analysis (simulation) of FWM characteristics, the total input pump and probe pulse, Vin(τ), is given by the following equation

$$V\_{in}(\tau) = V\_p(\tau) + V\_q(\tau) \exp(-i2\pi\Delta f \tau) \tag{16}$$

where, ( ) *Vp* τ and ( ) *Vq* τ are the complex envelope functions of the input pump and probe pulses respectively, ( /) *<sup>g</sup>* τ = −*t zv* is the local time that propagates with group velocity *<sup>g</sup> v* at the center frequency of an optical pulse, Δ*f* is the detuning frequency between the input pump and probe pulses and expressed as *<sup>p</sup> <sup>q</sup>* Δ= − *ff f* . Using the complex envelope function of (16), we solved the MNLSE and obtained the combined spectrum of the amplified pump, probe and the generated FWM signal at the output of SOA.

For the simulations, we used the parameters of a bulk SOA (AlGaAs/GaAs, double heterostructure) at a wavelength of 0.86 μm. The parameters are listed in Table 1 (Hong *et al.,* 1996; Das *et al.,* 2000). The length of the SOA was assumed to be 350 μm. All the results were obtained for a propagation step Δz of 5 μm. We confirmed that for any step size less than 5 μm the simulation results were almost identical (i.e., independent of the step size).

**Figure 5.** A simple schematic diagram for the simulation of FWM characteristics between pump and probe pulses in SOAs. The input pump and probe pulses with the center frequency of *pf* and *<sup>q</sup> f* are injected into the SOA. The pump and probe pulse detuning is Δ*f* . The FWM signal is generated at the output of the SOA.


**Table 1.** Simulation parameters of a bulk SOA (AlGaAs/GaAs, double heterostructure) (Hong *et al.,* 1996; Das *et al.,* 2000).

For the simulation of optical phase-conjugate characteristics in SOAs, we used the above model (Fig. 5). Fig. 5 shows a simple schematic diagram illustrating the simulation of the FWM characteristics in an SOA between short optical pulses. In SOA, the FWM signal is generated by mixing between the input pump and probe pulses, whose frequency appears at the symmetry position of the probe pulse with respect to the pump. For our simulation (as shown in Fig. 7), we have selected the detuning frequency between the input pump and probe pulses to +3 THz. The generated FWM signal is filtered using an optical narrow bandpass filter from the optical output spectrum containing the pump and probe signal. Here, the pass-band of the filter is set to be from +2 THz to +4 THz, i.e., a bandwidth of 2 THz is used. The shape of the pass-band was assumed to be rectangular.

#### **4. Optical phase-conjugation of picosecond FWM signals in SOAs**

96 Optical Communication

CH gain reduction

Name of the Parameters Symbols Values Units Length of SOA L 350 μm Effective area A 5 μm2 Center frequency of the pulse f0 349 THz Linear gain g0 92 cm-1 Group velocity dispersion β2 0.05 ps2 cm-1

Saturation energy Ws 80 pJ

The contribution of TPA h2 126 fs cm-1pJ-2

Instantaneous nonlinear Kerr effect n2 -0.70 cm2 TW-1 TPA coefficient γ2p 1.1 cm-1 W-1

**Table 1.** Simulation parameters of a bulk SOA (AlGaAs/GaAs, double heterostructure) (Hong *et al.,*

THz is used. The shape of the pass-band was assumed to be rectangular.

For the simulation of optical phase-conjugate characteristics in SOAs, we used the above model (Fig. 5). Fig. 5 shows a simple schematic diagram illustrating the simulation of the FWM characteristics in an SOA between short optical pulses. In SOA, the FWM signal is generated by mixing between the input pump and probe pulses, whose frequency appears at the symmetry position of the probe pulse with respect to the pump. For our simulation (as shown in Fig. 7), we have selected the detuning frequency between the input pump and probe pulses to +3 THz. The generated FWM signal is filtered using an optical narrow bandpass filter from the optical output spectrum containing the pump and probe signal. Here, the pass-band of the filter is set to be from +2 THz to +4 THz, i.e., a bandwidth of 2

A1 B1 A2 B2

Carrier lifetime τs 200 ps CH relaxation time τch 700 fs SHB relaxation time τshb 60 fs SHB saturation power Pshb 28.3 W Linear loss γ 11.5 cm-1

h1 0.13 cm-1pJ-1

0.15 -80 -60 0

fs μm-1 fs fs2 μm-1 fs2

Linewidth enhancement factor due to the CD αN 3.1 Linewidth enhancement factor due to the CH αT 2.0

The contribution of stimulated emission and FCA to the

Parameters describing second-order Taylor expansion

of the dynamically gain spectrum

1996; Das *et al.,* 2000).

#### **4.1. Chirp of FWM signal pulses for Fourier transform-limited input pulses**

In this sub-section, first, we obtain the frequency shift characteristics of FWM signal pulses for Fourier transform-limited input pulses. Fig. 6(a) shows the normalized output power of the pump, probe and FWM signal pulses. Fig. 6(b) shows the center frequency shifts of these pulses. Here, the input pump and probe pulses are Fourier transform limited (non-chirped) Gaussian pulses with a pulsewidth of 2 ps (full width at half maximum: FWHM). The pumprobe detuning is +3 THz. The input pump and probe pulse energy levels are 1 pJ and 0.1 pJ, respectively. The solid, dotted, and dotted-broken curves represent the pulse waveforms of the pump, probe, and generated FWM signal pulses, respectively. We have obtained these output waveforms by the method that have reported by Das et al., (Das et al., 2000). The output waveforms of the pump and probe pulses are close to the input waveforms. However, the peak position of the pump pulse is slightly shifted toward the leading edge due to the gain saturation of teh SOA (Shtaif & Eisenstein, 1995; Das et al., 2000). This is because the pulses have a larger gain at the leading edge than at the trailing edge. The pulsewidth of the FWM signal becomes narrower than that of the pump and probe pulses, because the FWM signal intensity is proportional to <sup>2</sup> *<sup>p</sup> <sup>q</sup> I I* , were *pI* is the pump pulse intensity and *qI* is the probe pulse intensity (Das et al., 2000).

The frequency shift characteristics of the pump, probe, and FWM signal pulses are shown in Fig. 6(b). The frequency shift of the FWM signal pulse is plotted for output power, which is greater than 1% of the output peak power. The vertical axis indicates the frequency shifts of the output pulses from the center frequencies of each input pulse. Two components of frequency shifts can be observed; negative frequency shift in the vicinity of the pulse peaks, and a frequency shift that is almost constant with time. In the vicinity of the pulse peaks, all the mixing pulses have negative pulse chirp, indicating that the pulse spectra are mainly red-shifted. A similar frequency shift was reported by Tang & Shore (Tang & Shore, 1999) for SOAs operating at a wavelength of 1.55 μm and attributed to the self- and cross- phase modulation under gain saturation. The origin of another frequency shift that is almost constant with time may be the effect of the gain spectrum of the SOA. The pump pulse frequency was set to near the gain peak. Therefore, both the higher and lower frequency components of the pump pulse have about the same optical gain. Thus the frequency of the pulse does not shift during the propagation in the SOA except for the frequency shift caused by the self- and cross- phase modulation as mentioned above. However, the probe pulse frequency was set at –3 THz from the center frequency of the pump pulse. Therefore, the higher frequency component of the probe pulse was enhanced more than the lower frequency component due to the gain spectrum, and the frequency of the output probe pulse is shifted toward the higher frequency side. This gain spectrum effect caused a +25 GHz shift in the probe pulses. The FWM signal is obtained at about +3 THz from the center frequency of the pump pulse. The lower frequency component of the FWM signal pulse was enhanced more than the higher frequency component. In addition, the probe pulse was shifted toward the higher frequency side. Therefore, the center frequency of the FWM signal pulse is shifted by about –35 GHz.

**Figure 6.** Normalized output power (a) and frequency shift (b) of the pump, probe, and generated FWM signal pulses. Input pump and probe pulses are Fourier transform-limited Gaussian shape with a pulsewidth of 2 ps (FWHM), detuning of the input pump and probe pulse is +3 THz. The input pump and probe energy levels are 1 pJ and 0.1 pJ, respectively.
