**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) used in the simulation, and nonlinear propagation of solitary pulses in SOAs.

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

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

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 high-

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.

FWM characteristics between the two input pump and probe pulses (Das *et al.,* 2007).

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

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) used in the simulation, and nonlinear propagation of

There are several types of "nonlinear effects" in SOAs. Among them, the important four "nonlinear effects" are shown in Fig. 1. These are (i) spectral hole-burning (SHB), (ii) carrier

heating (CH), (iii) carrier depletion (CD) and (iv) two-photon absorption (TPA).

speed optical communication systems.

**2. Analytical model** 

solitary pulses in SOAs.

**2.1 Important nonlinear effects in SOAs** 

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) 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. 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 (MNLSE).

#### **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:

$$\nabla^2 \overline{\mathbf{E}}(\mathbf{x}, y, z, \alpha) + \frac{\varepsilon\_r}{c^2} \alpha^2 \overline{\mathbf{E}}(\mathbf{x}, y, z, \alpha) = \mathbf{0} \tag{1}$$

Optical Demultiplexing Based on Four-Wave Mixing in Semiconductor Optical Amplifiers 169

<sup>1</sup> / <sup>2</sup> () 1 () ( ) *shb <sup>s</sup>*

( ) ( ) (1 ) ( )

*s s g h u s e e V s ds*

where, *gT*() is the resulting gain change due to the CH and TPA, *u*(s) is the unit step

carrier absorption to the CH gain reduction and *h2* is the contribution of two-photon

The dynamically varying slope and curvature of the gain plays a shaping role for pulses in the sub-picosecond range. The first and second order differential net (saturated) gain terms

(, ) (, ) *<sup>g</sup> A Bg g*

2 2 20 0 (, ) (, ) *<sup>g</sup> A Bg g*

> 

0 2

00 0 ( , ) ( , )/ ( ) ( , ) *N T g g fg*

*B1* and *B2* are constants describing changes in *A1* and *A2* with saturation, as given in (7)

The gain spectrum of an SOA is approximated by the following second-order Taylor

(, ) ( ) (, ) (, ) (, ) <sup>2</sup> *g g g g*

   

0

0

 

 

 

where, *A1* and *A2* are the slope and curvature of the linear gain at

2

 

> 

*f u s e V s ds*

) is given by (Hong *et al.,* 1996)

1

2

*T*

*shb shb*

) is the SHB function, *Pshb* is the SHB saturation power,

The resulting gain change due to the CH and TPA is given by (Hong *et al.,* 1996)

*P*

*)* is the saturated gain due to CD, *g0* is the linear gain, *Ws* is the saturation energy,

 

*<sup>T</sup>* are the linewidth enhancement factor associated with the gain changes

/ / 2

 

> 

 (9)

<sup>0</sup> <sup>0</sup>

2 2

 

(7)

(8)

*<sup>0</sup>*, respectively, while

(10)

1 10 0

 

(6)

 

/ / 4

 

( ) (1 ) ( )

*ch* is the CH relaxation time, *h1* is the contribution of stimulated emission and free-

*ch shb*

*h u s e e V s ds*

*s s*

*ch shb*

(5)

*shb* is the SHB relaxation

where, *gN(*

where, *f*(

time, and

function,

absorption.

and (8).

expansion in :

are (Hong *et al.,* 1996),

The SHB function *f*(

*N* and 

due to the CD and CH.

*<sup>s</sup>* is the carrier lifetime.

where, *Exyz* (,,, ) is the electromagnetic field of the pulse in the frequency domain, *c* is the velocity of light in vacuum and *<sup>r</sup>* is the nonlinear dielectric constant which is dependent on the electric field in a complex form. By slowly varying the envelope approximation and integrating the transverse dimensions we arrive at the pulse propagation equation in SOAs (Agrawal & Olsson, 1989; Dienes *et al.,* 1996).

$$\frac{\partial V(o, z)}{\partial z} = -i \left\{ \frac{o}{c} [1 + \chi\_n(o) + \Gamma \tilde{\chi}(o, N)]^{\frac{1}{2}} - \beta\_o \right\} V(o, z) \tag{2}$$

where, *V z* ( ,) is the Fourier-transform of *Vtz* (, ) representing pulse envelope, *<sup>m</sup>*( ) is the background (mode and material) susceptibility, ( ) is the complex susceptibility which represents the contribution of the active medium, *N* is the effective population density, *<sup>0</sup>* is the propagation constant. The quantity represents the overlap/ confinement factor of the transverse field distribution of the signal with the active region as defined in (Agrawal & Olsson, 1989).

Using mathematical manipulations (Sauter, 1996; Dienes *et al.,* 1996), including the real part of the instantaneous nonlinear Kerr effect as a single nonlinear index n2 and by adding the two-photon absorption (TPA) term we obtain the MNLSE for the phenomenological model of semiconductor laser and amplifiers (Hong *et al.,* 1996). The following MNLSE (Hong *et al.,* 1996; Das *et al.,* 2000) is used for the simulation of FWM characteristics with solitary probe pulse and optical DEMUX characteristics with multi-probe or pump in SOAs:

$$\begin{aligned} & \left[ \frac{\partial}{\partial z} - \frac{i}{2} \beta\_2 \frac{\partial^2}{\partial \tau^2} + \frac{\chi}{2} + \left( \frac{\chi\_{2p}}{2} + ib\_2 \right) \middle| V(\tau, z) \right]^2 V(\tau, z) \\ &= \left[ \frac{1}{2} g\_N(\tau) \middle| \frac{1}{f(\tau)} + ia\_N \right] + \frac{1}{2} \Delta g\_I(\tau) (1 + ia\_\Gamma) - i \frac{1}{2} \frac{\partial g(\tau, \phi)}{\partial \phi} \bigg|\_{\phi\_0} \frac{\partial}{\partial \tau} - \frac{1}{4} \frac{\partial^2 g(\tau, \phi)}{\partial \phi^2} \bigg|\_{\phi\_0} \frac{\partial^2}{\partial \tau^2} \right] V(\tau, z) \end{aligned} \tag{3}$$

We introduce the frame of local time (=*t* - z/v*g*), which propagates with a group velocity v*g* at the center frequency of an optical pulse. A slowly varying envelope approximation is used in (3), where the temporal variation of the complex envelope function is very slow compared with the cycle of the optical field. In (3), *V z* (,) is the time domain complex envelope function of an optical pulse, <sup>2</sup> *V z* (,) corresponding to the optical power, and 2 is the GVD. is the linear loss, *2p* is the two-photon absorption coefficient, *b*2 (= *0n2/cA*) is the instantaneous self-phase modulation term due to the instantaneous nonlinear Kerr effect *n2*, *0 (= 2f0)* is the center angular frequency of the pulse, *c* is the velocity of light in vacuum, *A (= wd/)* is the effective area (*d* and *w* are the thickness and width of the active region, respectively and is the confinement factor) of the active region.

The saturation of the gain due to the CD is given by (Hong *et al.,* 1996)

$$\mathcal{g}\_N(\tau) = \mathcal{g}\_0 \exp\left(-\frac{1}{W\_s} \int\_{-\infty}^{\tau} e^{-s/\tau\_s} \left| V(s) \right|^2 ds\right) \tag{4}$$

where, *gN()* is the saturated gain due to CD, *g0* is the linear gain, *Ws* is the saturation energy, *<sup>s</sup>* is the carrier lifetime.

The SHB function *f*() is given by (Hong *et al.,* 1996)

168 Optical Communications Systems

on the electric field in a complex form. By slowly varying the envelope approximation and integrating the transverse dimensions we arrive at the pulse propagation equation in SOAs

<sup>0</sup>

*<sup>0</sup>* is the propagation constant. The quantity represents the overlap/ confinement

 

<sup>2</sup> ( ,) 1 () (, ) ( ,) *<sup>m</sup> V z <sup>i</sup> <sup>N</sup> V z*

is the Fourier-transform of *Vtz* (, ) representing pulse envelope,

which represents the contribution of the active medium, *N* is the effective population

factor of the transverse field distribution of the signal with the active region as defined in

Using mathematical manipulations (Sauter, 1996; Dienes *et al.,* 1996), including the real part of the instantaneous nonlinear Kerr effect as a single nonlinear index n2 and by adding the two-photon absorption (TPA) term we obtain the MNLSE for the phenomenological model of semiconductor laser and amplifiers (Hong *et al.,* 1996). The following MNLSE (Hong *et al.,* 1996; Das *et al.,* 2000) is used for the simulation of FWM characteristics with solitary probe

11 1 1 1 (, ) (, ) ( ) ( )(1 ) (,) 2 () 2 <sup>2</sup> <sup>4</sup>

velocity v*g* at the center frequency of an optical pulse. A slowly varying envelope approximation is used in (3), where the temporal variation of the complex envelope

thickness and width of the active region, respectively and is the confinement factor) of

<sup>1</sup> ( ) exp ( ) *<sup>s</sup> <sup>s</sup>*

*s g g e V s ds W*

 

is the linear loss,

 

function is very slow compared with the cycle of the optical field. In (3), *V z* (,)

*0 (= 2*

*g g <sup>g</sup> <sup>i</sup> <sup>g</sup> i i V z*

 

> 

 

pulse and optical DEMUX characteristics with multi-probe or pump in SOAs:

time domain complex envelope function of an optical pulse, <sup>2</sup> *V z* (,)

The saturation of the gain due to the CD is given by (Hong *et al.,* 1996)

0

2 is the GVD.

2 2

 

(,) (,) 2 22

*<sup>i</sup> ib V z V z*

*p*

*N NT T*

pulse, *c* is the velocity of light in vacuum, *A (= wd/*

*N*

 

*z c*

the background (mode and material) susceptibility,

is the electromagnetic field of the pulse in the frequency domain, *c* is the

1

(2)

( ) 

<sup>0</sup> <sup>0</sup>

(=*t* - z/v*g*), which propagates with a group

(4)

*f0)* is the center angular frequency of the

*)* is the effective area (*d* and *w* are the

*0n2/cA*) is the instantaneous self-phase modulation term due to the

/ 2

 2 2 2 2

 

*2p* is the two-photon absorption

 

corresponding to

(3)

is the

 *<sup>m</sup>*( ) is

is the complex susceptibility

is the nonlinear dielectric constant which is dependent

where, *Exyz* (,,, )

where, *V z* ( ,) 

(Agrawal & Olsson, 1989).

2

the optical power, and

coefficient, *b*2 (=

the active region.

*f*

2 2 2

 

We introduce the frame of local time

instantaneous nonlinear Kerr effect *n2*,

density,

*z*

velocity of light in vacuum and *<sup>r</sup>*

(Agrawal & Olsson, 1989; Dienes *et al.,* 1996).

$$f(\tau) = 1 + \frac{1}{\tau\_{sbb} P\_{sbb}} \int\_{-\infty}^{+\infty} \mu(s) e^{-s/\tau\_{sbb}} \left| V(\tau - s) \right|^2 ds \tag{5}$$

where, *f*() is the SHB function, *Pshb* is the SHB saturation power, *shb* is the SHB relaxation time, and *N* and *<sup>T</sup>* are the linewidth enhancement factor associated with the gain changes due to the CD and CH.

The resulting gain change due to the CH and TPA is given by (Hong *et al.,* 1996)

$$\begin{split} \Lambda g\_T(\tau) &= -h\_1 \int\_{-\infty}^{+\infty} \mu(s) e^{-s/\tau\_{ch}} \left( 1 - e^{-s/\tau\_{sh}} \right) \left| V(\tau - s) \right|^2 ds \\ &- h\_2 \int\_{-\infty}^{+\infty} \mu(s) e^{-s/\tau\_{ch}} \left( 1 - e^{-s/\tau\_{sh}} \right) \left| V(\tau - s) \right|^4 ds \end{split} \tag{6}$$

where, *gT*() is the resulting gain change due to the CH and TPA, *u*(s) is the unit step function, *ch* is the CH relaxation time, *h1* is the contribution of stimulated emission and freecarrier absorption to the CH gain reduction and *h2* is the contribution of two-photon absorption.

The dynamically varying slope and curvature of the gain plays a shaping role for pulses in the sub-picosecond range. The first and second order differential net (saturated) gain terms are (Hong *et al.,* 1996),

$$\left.\frac{\partial \mathbf{g}(\tau, o\boldsymbol{\varrho})}{\partial o}\right|\_{o\_0} = A\_1 + B\_1 \left[\mathbf{g}\_0 - \mathbf{g}(\tau, o\boldsymbol{\varrho}\_0)\right] \tag{7}$$

$$\left. \frac{\partial^2 g(\tau, \alpha)}{\partial \alpha^2} \right|\_{\alpha\_0} = A\_2 + B\_2 \left[ g\_0 - g(\tau, \alpha\_0) \right] \tag{8}$$

$$\mathcal{g}(\tau, \alpha\_0) = \mathcal{g}\_N(\tau, \alpha\_0) / \left/ f(\tau) + \Delta \mathcal{g}\_T(\tau, \alpha\_0) \right. \tag{9}$$

where, *A1* and *A2* are the slope and curvature of the linear gain at *<sup>0</sup>*, respectively, while *B1* and *B2* are constants describing changes in *A1* and *A2* with saturation, as given in (7) and (8).

The gain spectrum of an SOA is approximated by the following second-order Taylor expansion in :

$$\log(\tau,\alpha) = \mathcal{g}(\tau,\alpha\_0) + \Delta\alpha \frac{\partial \mathcal{g}(\tau,\alpha)}{\partial \alpha} \bigg|\_{\alpha\_0} + \frac{\left(\Delta\alpha\right)^2}{2} \frac{\partial^2 \mathcal{g}(\tau,\alpha)}{\partial \alpha^2} \bigg|\_{\alpha\_0} \tag{10}$$

Optical Demultiplexing Based on Four-Wave Mixing in Semiconductor Optical Amplifiers 171

approximated by the second-order Taylor expression series. As the pulse propagates in the SOA, the pulse intensity increases due to the gain of the SOA. The increase in pulse intensity reduces the gain, and the center frequency of the gain shifts to lower frequencies. The pump

set -3 THz from for the calculations of FWM characteristics as described below, and the linear gain *g*0 is -42 cm at this frequency. Although the probe frequency lies outside the gain bandwidth, we selected a detuning of 3 THz in this simulation because the FWM signal must be spectrally separated from the output of the SOA. As will be shown later, even for this large degree of detuning, the FWM signal pulse and the pump pulse spectrally overlap when the pulsewidths become short (<0.5 ps) (Das *et al.,* 2001). The gain bandwidth is about the same as the measured value for an AlGaAs/GaAs bulk SOA (Seki *et al.,* 1981). If an InGaAsP/InP bulk SOA is used we can expect much wider gain bandwidth (Leuthold *et al.,* 2000). With a decrease in the carrier density, the gain decreases and the peak position is shifted to a lower frequency because of the band-filling effect. Figure 2(b) shows the saturated gain versus input pump pulse energy characteristics of the SOA. When the input pump pulsewidth decreases then the small signal gain decreases due to the spectral limit of the gain bandwidth. For the case, when the input pump pulsewidth is short (very narrow, such as 200 fs or lower), the gain saturates at small input pulse energy (Das *et al.,* 2000). This is due to the CH and SHB with the fast response. Initially, the MNLSE was used by (Hong *et al.,* 1996) for the analysis of "solitary pulse" propagation in an SOA. We used the same MNLSE for the simulation of FWM and optical DEMUX characteristics in SOA using the FD-BPM. Here, we have introduced a complex envelope function V(, 0) at the input side of the SOA for taking into account the two (pump

To solve a boundary value problem using the finite-differences method, every derivative term appearing in the equation, as well as in the boundary conditions, is replaced by the central differences approximation. Central differences are usually preferred because they lead to an excellent accuracy (Conte & Boor, 1980). In the modeling, we used the finite-

Usually, the fast Fourier transformation beam propagation method (FFT-BPM) (Okamoto, 1992; Brigham, 1988) is used for the analysis of the optical pulse propagation in optical fibers by the successive iterations of the Fourier transformation and the inverse Fourier transformation. In the FFT-BPM, the linear propagation term (GVD term) and phase compensation terms (other than GVD, 1st and 2nd order gain spectrum terms) are separated in the nonlinear Schrödinger equation for the individual consideration of the time and frequency domain for the optical pulse propagation. However, in our model, equation (3) includes the dynamic gain change terms, i.e., the 1st and 2nd order gain spectrum terms which are the last two terms of the right-side in equation (3). Therefore, it is not possible to separate equation (3) into the linear propagation term and phase compensation term and it is quite difficult to calculate equation (3) using the FFT-BPM. For this reason, we used the FD-BPM (Chung & Dagli, 1990; Conte & Boor, 1980; Das *et al.,* 2000; Razaghi *et al.,* 2009). If we replace the time derivative terms of equation (3) by the below central-difference approximation, equation (11), and integrate equation (3) with the small propagation step z,

0. The probe frequency is

frequency is set to near the gain peak, and linear gain *g*0 is 92 cm at

and probe) or more (multi-pump or probe) pulses.

**2.3 Finite-difference beam propagation method (FD-BPM)** 

we obtain the tridiagonal simultaneous matrix equation (12)

differences (central differences) to solve the MNLSE for this analysis.

The coefficients 0 *g*(, ) and 0 2 2 *g*(, ) are related to *A1*, *B1*, *A2* and *B2* by (7) and (8).

Here we assumed the same values of *A1*, *B1*, *A2* and *B2* as in (Hong *et al.,* 1996) for an AlGaAs/GaAs bulk SOA.

The time derivative terms in (3) have been replaced by the central-difference approximation in order to simulate this equation by the FD-BPM (Das *et al.,* 2000). In simulation, the parameter of bulk SOAs (AlGaAs/GaAs, double heterostructure) with a wavelength of 0.86 m (Hong *et al.,* 1996) is used and the SOA length is 350 m. The input pulse shape is sech2 and is Fourier transform-limited.

Fig. 2. (a) The gain spectra given by the second-order Taylor expansion about the center frequency of the pump pulse 0. The solid line shows the unsaturated gain spectrum (length: 0 m), the dotted and the dashed-dotted lines are a saturated gain spectrum at 175 m and 350 m, respectively. Here, the input pump pulse pulsewidth is 1 ps and pulse energy is 1 pJ. (b) Saturated gain versus the input pump pulse energy characteristics of the SOA. The saturation energy decreases with decreasing the input pump pulsewidth. The SOA length is 350 m. The input pulsewidths are 0.2 ps, 0.5 ps, and 1 ps respectively, and a pulse energy of 1 pJ.

The gain spectra of SOAs are very important for obtaining the propagation and wave mixing (FWM and optical DEMUX between the input pump and probe pulses) characteristics of short optical pulses. Figure 2(a) shows the gain spectra given by a second-order Taylor expansion about the pump pulse center frequency *0* with derivatives of *g(, )* by (7) and (8) (Das *et al.,*  2000). In Fig. 2(a), the solid line represents an unsaturated gain spectrum (length: 0 m), the dotted line represents a saturated gain spectrum at the center position of the SOA (length: 175 m), and the dashed–dotted line represents a saturated gain spectrum at the output end of the SOA (length: 350m), when the pump pulsewidth is 1 ps and input energy is 1 pJ. These gain spectra were calculated using (1), because, the waveforms of optical pulses depend on the propagation distance (i.e., the SOA length). The spectra of these pulses were obtained by Fourier transformation. The "local" gains at the center frequency at *z* = 0, 175, and 350 m were obtained from the changes in the pulse intensities at the center frequency at around those positions (Das *et al.,* 2001). The gain at the center frequency in the gain spectrum was

Here we assumed the same values of *A1*, *B1*, *A2* and *B2* as in (Hong *et al.,* 1996) for an

The time derivative terms in (3) have been replaced by the central-difference approximation in order to simulate this equation by the FD-BPM (Das *et al.,* 2000). In simulation, the parameter of bulk SOAs (AlGaAs/GaAs, double heterostructure) with a wavelength of 0.86 m (Hong *et al.,* 1996) is used and the SOA length is 350 m. The input pulse shape is sech2

0

5

10

Saturated Gain (dB)

15

(b)

are related to *A1*, *B1*, *A2* and *B2* by (7) and (8).

0.001 0.01 0.1 1 10

0. The solid line shows the unsaturated gain spectrum (length: 0

*,* 

Input Energy (pJ)

*)* by (7) and (8) (Das *et al.,* 

Pump Pulsewidth = 0.2 ps

0.5 ps 1 ps

0

175 m 350 m

Fig. 2. (a) The gain spectra given by the second-order Taylor expansion about the center

input pulsewidths are 0.2 ps, 0.5 ps, and 1 ps respectively, and a pulse energy of 1 pJ.

m), the dotted and the dashed-dotted lines are a saturated gain spectrum at 175 m and 350 m, respectively. Here, the input pump pulse pulsewidth is 1 ps and pulse energy is 1 pJ. (b) Saturated gain versus the input pump pulse energy characteristics of the SOA. The saturation energy decreases with decreasing the input pump pulsewidth. The SOA length is 350 m. The

The gain spectra of SOAs are very important for obtaining the propagation and wave mixing (FWM and optical DEMUX between the input pump and probe pulses) characteristics of short optical pulses. Figure 2(a) shows the gain spectra given by a second-order Taylor expansion

2000). In Fig. 2(a), the solid line represents an unsaturated gain spectrum (length: 0 m), the dotted line represents a saturated gain spectrum at the center position of the SOA (length: 175 m), and the dashed–dotted line represents a saturated gain spectrum at the output end of the SOA (length: 350m), when the pump pulsewidth is 1 ps and input energy is 1 pJ. These gain spectra were calculated using (1), because, the waveforms of optical pulses depend on the propagation distance (i.e., the SOA length). The spectra of these pulses were obtained by Fourier transformation. The "local" gains at the center frequency at *z* = 0, 175, and 350 m were obtained from the changes in the pulse intensities at the center frequency at around those positions (Das *et al.,* 2001). The gain at the center frequency in the gain spectrum was

*0* with derivatives of *g(*

<sup>0</sup> Length =

The coefficients



Gain, g (c

m-1)

0

50

(a)

Probe

frequency of the pump pulse

about the pump pulse center frequency

100

0


Pump ()

Frequency (THz)

and

2

 

2 *g*(, )

 

*g*(, )

 

 

and is Fourier transform-limited.

AlGaAs/GaAs bulk SOA.

approximated by the second-order Taylor expression series. As the pulse propagates in the SOA, the pulse intensity increases due to the gain of the SOA. The increase in pulse intensity reduces the gain, and the center frequency of the gain shifts to lower frequencies. The pump frequency is set to near the gain peak, and linear gain *g*0 is 92 cm at 0. The probe frequency is set -3 THz from for the calculations of FWM characteristics as described below, and the linear gain *g*0 is -42 cm at this frequency. Although the probe frequency lies outside the gain bandwidth, we selected a detuning of 3 THz in this simulation because the FWM signal must be spectrally separated from the output of the SOA. As will be shown later, even for this large degree of detuning, the FWM signal pulse and the pump pulse spectrally overlap when the pulsewidths become short (<0.5 ps) (Das *et al.,* 2001). The gain bandwidth is about the same as the measured value for an AlGaAs/GaAs bulk SOA (Seki *et al.,* 1981). If an InGaAsP/InP bulk SOA is used we can expect much wider gain bandwidth (Leuthold *et al.,* 2000). With a decrease in the carrier density, the gain decreases and the peak position is shifted to a lower frequency because of the band-filling effect. Figure 2(b) shows the saturated gain versus input pump pulse energy characteristics of the SOA. When the input pump pulsewidth decreases then the small signal gain decreases due to the spectral limit of the gain bandwidth. For the case, when the input pump pulsewidth is short (very narrow, such as 200 fs or lower), the gain saturates at small input pulse energy (Das *et al.,* 2000). This is due to the CH and SHB with the fast response.

Initially, the MNLSE was used by (Hong *et al.,* 1996) for the analysis of "solitary pulse" propagation in an SOA. We used the same MNLSE for the simulation of FWM and optical DEMUX characteristics in SOA using the FD-BPM. Here, we have introduced a complex envelope function V(, 0) at the input side of the SOA for taking into account the two (pump and probe) or more (multi-pump or probe) pulses.
