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

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

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, we obtain the tridiagonal simultaneous matrix equation (12)

Optical Demultiplexing Based on Four-Wave Mixing in Semiconductor Optical Amplifiers 173

*|V(, z)|2*

Input Pulse

.

*|V(, z)|2*

.

z: Propagation Step

*tzv* is

0

z

0

Fig. 3. A simple schematic diagram of FD-BPM in the time domain, where, ( / ) *<sup>g</sup>*

the local time, which propagates with the group velocity *<sup>g</sup> v* at the center frequency of an optical pulse and is the sampling time, and *z* is the propagation direction and z is the

The FD-BPM (Conte & Boor, 1980; Chung & Dagli, 1990; Das *et al.,* 2000; Razaghi *et al.,* 2009a & 2009b) is used for the simulation of several important characteristics, namely, (1) single pulse propagation in SOAs (Das *et al.,* 2008; Razaghi *et al.,* 2009a & 2009b), (2) two input pulses propagating in SOAs (Das *et al.,* 2000; Connelly *et al.,* 2008), (3) Optical DEMUX characteristics of multi-probe or pump input pulses based on FWM in SOAs (Das *et al.,* 2001), (4) Optical phase-conjugation characteristics of picosecond FWM signal in SOAs (Das *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, self-phase modulation (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

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

z+z

. . .

*, 0)|2*

. *|V(*

.

z

Propagation with Group Velocity vg

Power

0

propagation step.

0 . i i+1 

Sampling Time:

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

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

temperature in the propagation equation (i.e., MNLSE).

$$\frac{\partial}{\partial \tau} V\_k = \frac{V\_{k+1} - V\_{k-1}}{2\Delta \tau}, \quad \frac{\partial^2}{\partial \tau^2} V\_k = \frac{V\_{k+1} - 2V\_k + V\_{k-1}}{\Delta \tau^2} \tag{11}$$

where, ( ) *V V k k* , 1 ( ) *V V k k* , and 1 ( ) *V V k k* 

$$\begin{aligned} -a\_k(z + \Delta z) \, V\_{k-1}(z + \Delta z) + \left\{1 - b\_k(z + \Delta z)\right\} \, V\_k(z + \Delta z) - c\_k(z + \Delta z) \, V\_{k+1}(z + \Delta z) \\ = a\_k(z) \, V\_{k-1}(z) + \left\{1 + b\_k(z)\right\} \, V\_k(z) + c\_k(z) \, V\_{k+1}(z) \end{aligned} \tag{12}$$

where, *k n* 1, 2, 3, ............, and

$$a\_k(z) = \frac{\Delta z}{2} \left[ \frac{i\mathcal{B}\_2}{2\Delta\tau^2} + i \frac{1}{4\Delta\tau} \frac{\partial g(\tau, \rho, z)}{\partial \rho}|\_{\alpha\_0, \tau\_k} - \frac{1}{4\Delta\tau^2} \frac{\partial^2 g(\tau, \rho, z)}{\partial \rho^2}|\_{\alpha\_0, \tau\_k} \right] \tag{13}$$

$$\begin{split} b\_{k}(z) &= -\frac{\Delta z}{2} \left| \frac{i\beta\_{2}}{\Delta \tau^{2}} + \frac{\gamma}{2} + \left( \frac{\gamma\_{2p}}{2} + ib\_{2} \right) \right| V\_{k}(z) \left|^{2} - \frac{1}{2} g\_{N}(\tau\_{k}, a\_{0}, z) (1 + ia\_{N}) \right. \\ &\left. - \frac{1}{2} \Delta g\_{T}(\tau\_{k'}, a\_{0}, z) (1 + ia\_{T}) - \frac{1}{2\Delta \tau^{2}} \frac{\partial^{2} g(\tau, a\_{0}, z)}{\partial a^{2}} \right|\_{a\_{0}, \tau\_{k}} \end{split} \tag{14}$$

$$c\_k(z) = \frac{\Delta z}{2} \left[ \frac{i\beta\_2}{2\Lambda\tau^2} - i\frac{1}{4\Lambda\tau} \frac{\partial \mathbf{g}(\tau, \alpha, z)}{\partial \alpha}|\_{\alpha\_0, \tau\_k} - \frac{1}{4\Lambda\tau^2} \frac{\partial^2 \mathbf{g}(\tau, \alpha, z)}{\partial \alpha^2}|\_{\alpha\_0, \tau\_k} \right] \tag{15}$$

where, is the sampling time and *n* is the number of sampling. If we know ( ) *V z <sup>k</sup>* , ( *k n* 1, 2, 3, .........., ) at the position *z* , we can calculate ( ) *Vz z <sup>k</sup>* at the position of *z z* which is the propagation of a step *z* from position *z* , by using equation (12). It is not possible to directly calculate equation (12) because it is necessary to calculate the left-side terms ( ) *<sup>k</sup> az z* , ( ) *<sup>k</sup> bz z* , and ( ) *<sup>k</sup> cz z* of equation (12) from the unknown ( ) *Vz z <sup>k</sup>* . Therefore, we initially defined ( ) () *k k az z az* , ( ) () *k k bz z bz* , and ( ) () *k k cz z cz* and obtained (0)( ) *Vzz <sup>k</sup>* , as the zeroth order approximation of ( ) *Vz z <sup>k</sup>* by using equation (12). We then substituted (0)( ) *Vzz <sup>k</sup>* in equation (12) and obtained (1)( ) *Vzz <sup>k</sup>* as the first order approximation of ( ) *Vz z <sup>k</sup>* and finally obtained the accurate simulation results by the iteration as used in (Brigham, 1988; Chung & Dagli, 1990; Das *et al.,* 2000; Razaghi *et al.,* 2009).

Figure 3 shows a simple schematic diagram of the FD-BPM in time domain. Here, ( /) *<sup>g</sup> tzv* is the local time, which propagates with the group velocity *<sup>g</sup> v* at the center frequency of an optical pulse and is the sampling time. z is the propagation direction and z is the propagation step. With this procedure, we used up to 3-rd time iteration for more accuracy of the simulations.

2

*V V V VV V V*

*k k*

, and 1 ( ) *V V k k*

*kk k k kk*

 

 , 1 ( ) *V V k k* 

2

 

2

where, ( ) *V V k k*

where,

Razaghi *et al.,* 2009).

accuracy of the simulations.

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

where, *k n* 1, 2, 3, ............, and

<sup>2</sup> , <sup>2</sup> *k k k kk*

1 1

1 1 (, ,) (, ,) ( ) <sup>|</sup> <sup>|</sup> 2 4 2 4 *k k <sup>k</sup> z i gz gz a z <sup>i</sup>*

 

2 2 2

 

*p*

 

2

*z i b z ib V z g z i*

1 1 (, ,) ( , , )(1 ) <sup>|</sup> <sup>2</sup> <sup>2</sup> *<sup>k</sup>*

1 1 (, ,) (, ,) ( ) <sup>|</sup> <sup>|</sup> 2 4 2 4 *k k <sup>k</sup> z i gz gz c z <sup>i</sup>*

( *k n* 1, 2, 3, .........., ) at the position *z* , we can calculate ( ) *Vz z <sup>k</sup>* at the position of *z z* which is the propagation of a step *z* from position *z* , by using equation (12). It is not possible to directly calculate equation (12) because it is necessary to calculate the left-side terms ( ) *<sup>k</sup> az z* , ( ) *<sup>k</sup> bz z* , and ( ) *<sup>k</sup> cz z* of equation (12) from the unknown ( ) *Vz z <sup>k</sup>* . Therefore, we initially defined ( ) () *k k az z az* , ( ) () *k k bz z bz* , and ( ) () *k k cz z cz* and obtained (0)( ) *Vzz <sup>k</sup>* , as the zeroth order approximation of ( ) *Vz z <sup>k</sup>* by using equation (12). We then substituted (0)( ) *Vzz <sup>k</sup>* in equation (12) and obtained (1)( ) *Vzz <sup>k</sup>* as the first order approximation of ( ) *Vz z <sup>k</sup>* and finally obtained the accurate simulation results by the iteration as used in (Brigham, 1988; Chung & Dagli, 1990; Das *et al.,* 2000;

Figure 3 shows a simple schematic diagram of the FD-BPM in time domain. Here,

 *tzv* is the local time, which propagates with the group velocity *<sup>g</sup> v* at the center frequency of an optical pulse and is the sampling time. z is the propagation direction and z is the propagation step. With this procedure, we used up to 3-rd time iteration for more

*g z g zi*

 

> 

2 0

<sup>1</sup> ( ) | ( )| ( , , )(1 ) 2 22 <sup>2</sup>

*k k Nk N*

0 , 2 2

 

0 0

*T k T*

( ) ( )1 ( ) ( ) ( ) ( )

*a z zV z z b z z V z z c z zV z z*

1 1 1 1 2 2

2 , , 2 2

2

2 , , 2 2

is the sampling time and *n* is the number of sampling. If we know ( ) *V z <sup>k</sup>* ,

 

 

> 

 

 

 

*kk k k kk*

1 1

 

0

 

 

2

(12)

 

> 

  (13)

(14)

(15)

() () 1 () () () ()

*a zV z b z V z c zV z*

0 0 2

 

(11)

Fig. 3. A simple schematic diagram of FD-BPM in the time domain, where, ( / ) *<sup>g</sup> tzv* is the local time, which propagates with the group velocity *<sup>g</sup> v* at the center frequency of an optical pulse and is the sampling time, and *z* is the propagation direction and z is the propagation step.

The FD-BPM (Conte & Boor, 1980; Chung & Dagli, 1990; Das *et al.,* 2000; Razaghi *et al.,* 2009a & 2009b) is used for the simulation of several important characteristics, namely, (1) single pulse propagation in SOAs (Das *et al.,* 2008; Razaghi *et al.,* 2009a & 2009b), (2) two input pulses propagating in SOAs (Das *et al.,* 2000; Connelly *et al.,* 2008), (3) Optical DEMUX characteristics of multi-probe or pump input pulses based on FWM in SOAs (Das *et al.,* 2001), (4) Optical phase-conjugation characteristics of picosecond FWM signal in SOAs (Das *et al.,* 2001), and (5) FWM conversion efficiency with optimum time-delays between the input pump and probe pulses (Das *et al.,* 2007).

#### **2.4 Nonlinear optical 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, self-phase modulation (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).

Optical Demultiplexing Based on Four-Wave Mixing in Semiconductor Optical Amplifiers 175

h1 0.13 cm-1pJ-1

0.15 -80 -60 0

fs m-1 fs fs2 m-1 fs2

A1 B1 A2 B2

Name of the Parameters Symbols Values Units Length of SOA L 350 m Effective area A 5 m2 Center frequency of the pulse f0 349 THz Linear gain g0 92 cm-1 Group velocity dispersion <sup>2</sup> 0.05 ps2 cm-1 Saturation energy Ws 80 pJ

The contribution of TPA h2 126 fs cm-1pJ-2 Carrier lifetime <sup>s</sup> 200 ps CH relaxation time ch 700 fs SHB relaxation time shb 60 fs SHB saturation power Pshb 28.3 W Linear loss 11.5 cm-1 Instantaneous nonlinear Kerr effect n2 -0.70 cm2 TW-1 TPA coefficient 2p 1.1 cm-1 W-1

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

simulation results were almost identical (i.e., independent of the step size).

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

For the simulation of optical DEMUX characteristics in SOAs, we have started with the simulation of FWM characteristics for solitary probe pulse's. Figure 5 shows a simple schematic diagram illustrating the simulation of the FWM characteristics in an SOA between short optical pulses. In SOAs, 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. 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. The solid line represents for a short pump pulsewidths and the dotted line represents for a wider pump pulsewidths. For a wider pulsewidth, the pump peak intensity decreases, spectral peak intensity increases, and the FWM signal peak intensity decreases as shown in the figure

Linewidth enhancement factor due to the CD <sup>N</sup> 3.1 Linewidth enhancement factor due to the CH <sup>T</sup> 2.0

The contribution of stimulated emission and FCA to

Parameters describing second-order Taylor expansion

of the dynamically gain spectrum

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

when the input pulse energy is kept constant.

the CH gain reduction

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 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 DEMUX characteristics in SOAs with the multiinput pump and probe pulses.

Fig. 4. A simple schematic diagram for the simulation of nonlinear single pulse propagation in SOA. Here, <sup>2</sup> *V*( ,0) is the input (z = 0) pulse intensity and <sup>2</sup> *V z* (,) 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, is the local time, <sup>2</sup> *V*( ,0) is the intensity (power) of input pulse (z = 0) and <sup>2</sup> *V z* (,) 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 FWM (with single probe) and DEMUX (with multi-bit pump or probe pulses) characteristics in SOAs.

#### **3. FWM characteristics in SOAs with the solitary probe pulse**

In this section, we will discuss the FWM characteristics with the solitary probe pulse 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* and ( ) *Vq* are the complex envelope functions of the input pump and probe pulses respectively, ( /) *<sup>g</sup> tzv* is the local time that propagates with group velocity *<sup>g</sup> v* at the center frequency of an optical pulse, *f* is the detuning frequency between the input pump and probe pulses and expressed as *p q 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.

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 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 DEMUX characteristics in SOAs with the multi-

SOA

Fig. 4. A simple schematic diagram for the simulation of nonlinear single pulse propagation

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

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 SOA. We used this model to simulate FWM (with single probe) and DEMUX (with multi-bit

In this section, we will discuss the FWM characteristics with the solitary probe pulse 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

> 

at the center frequency of an optical pulse, *f* is the detuning frequency between the input pump and probe pulses and expressed as *p q ff f* . Using the complex envelope function of (16), we solved the MNLSE and obtained the combined spectrum of the amplified pump,

 

*tzv* is the local time that propagates with group velocity *<sup>g</sup> v*

are the complex envelope functions of the input pump and probe

*f* (16)

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

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

**3. FWM characteristics in SOAs with the solitary probe pulse** 

 ( ) ( ) ( )exp( 2 ) *V VV i in p q* 

probe and the generated FWM signal at the output of SOA.

*|V(, z)|2*

.

is the output pulse

is the

is the

0

Output Pulse

input pump and probe pulses.

in SOA. Here, <sup>2</sup> *V*( ,0)

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

equation

where, ( ) *Vp*

 and ( ) *Vq* 

pulses respectively, ( /) *<sup>g</sup>* 

.

Input Pulse

*|V(, 0)|2*

0

intensity (after propagating a distance z) of SOA.

pump or probe pulses) characteristics in SOAs.


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

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

For the simulation of optical DEMUX characteristics in SOAs, we have started with the simulation of FWM characteristics for solitary probe pulse's. Figure 5 shows a simple schematic diagram illustrating the simulation of the FWM characteristics in an SOA between short optical pulses. In SOAs, 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. 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. The solid line represents for a short pump pulsewidths and the dotted line represents for a wider pump pulsewidths. For a wider pulsewidth, the pump peak intensity decreases, spectral peak intensity increases, and the FWM signal peak intensity decreases as shown in the figure when the input pulse energy is kept constant.

Optical Demultiplexing Based on Four-Wave Mixing in Semiconductor Optical Amplifiers 177

0

0

15

0

O utput P ow er (mW )

(a) (b)

Fig. 6. (a) Output spectra of the SOA before filtering. The solid and dashed curves show the output spectra with pump and probe pulses and with only a pump pulse, respectively. (b) Output pulse waveforms after filtering from +2 ~ +4 THz. The solid and dashed curves show the output pulse waveforms with pump and probe pulses and with only a pump pulse, respectively. Here, the input pump and probe pulse energies are 1 pJ and 10 fJ,

5

10

O utput P ow er (mW )

0.1

0.2

0.3

0.5 ps

0.2 ps


Time (ps)

O utput P ow er (mW )

0.05

0.1

Pump Pulsewidth = 1 ps




Spectral Power (dBm/nm)

Spectral Power (dBm/nm)

Spectral Power (dBm/nm)

Pump Pulsewidth = 1 ps

0.5 ps

0.2 ps


respectively. The input probe pulsewidth is 1 ps.

Frequency (THz)

Probe Pump FWM Signal

Filter bandwidth

Fig. 5. A simple schematic diagram for the simulation of FWM characteristics for solitary probe pulse for the optimization of the input pump pulsewidth. Here, the input pump pulsewidth is varied.

Figure 6(a) shows the calculated output spectra of the SOA. The solid and dashed curves show the output spectra with pump and probe pulses and with only a pump pulse, respectively. The pump pulsewidths are 1 ps, 0.5 ps, and 0.2 ps and the probe pulsewidth is 1 ps. The input pump and probe pulse energies are 1 pJ and 10 fJ, respectively. For a pump pulsewidth of 1 ps, the spectral peaks of the pump, probe, and FWM signals are clearly separated. The FWM signal can be obtained by the spectral filtering whose bandwidth is from +2 THz to +4 THz, which is shown in the figure by the arrow. With the decrease in the pump pulsewidth, the pump spectral width is broadened and it becomes difficult to extract the FWM signal using the optical filter due to the spectral overlap. For the shorter pump pulsewidth less than 0.5 ps, the FWM signals are not clearly observed.

Figure 6(b) shows the temporal waveforms of the output signals after filtering. The solid and dashed curves show the waveforms with probe and without probe pulses, respectively. The contrast between the output power with probe and without probe pulses for a pump pulsewidth of 1 ps is larger than that for the shorter pump pulsewidths. This is due to the strong overlap between pump pulse of 0.5 ps and 0.2 ps and the FWM signal in the frequency domain. For an input pump pulsewidth of 1 ps, a FWM signal pulsewidth of 0.73 ps is narrower than the input pump pulsewidth. This is due to the fact that the FWM signal intensity is proportional to <sup>2</sup> *p q I I* i.e., <sup>2</sup> *FWM p q I II* as reported (Das *et al.,* 2000). Here, *IFWM* is the FWM signal intensity, *pI* is the pump pulse intensity, and *qI* is the probe pulse intensity. For input pump pulsewidths of 0.5 ps and 0.2 ps, the optical bandpass filter broadens the FWM signal pulsewidth due to the limitation in the frequency domain. Then, FWM signal pulsewidths of 0.57 ps and 0.55 ps become broader than the input pump pulsewidths. By this filtering, the energies of the transform-limited sech2 pulses with pulsewidths of 1 ps, 0.73 ps, 0.5 ps, and 0.2 ps are reduced by 0.002%, 0.05%, 0.7%, and 19%, respectively. The peak powers are decreased by 0.86%, 4%, 14%, and 63%, respectively. The pulsewidth of 0.73 ps corresponds the that of the FWM pulse among 1 ps pump and 1 ps probe pulses (Fig. 6(b)). Therefore, the waveform distortion by this filtering is negligibly small for the FWM pulses among 1 ps pump and 1 ps probe pulses.

SOA **~~~**

Fig. 5. A simple schematic diagram for the simulation of FWM characteristics for solitary probe pulse for the optimization of the input pump pulsewidth. Here, the input pump

Figure 6(a) shows the calculated output spectra of the SOA. The solid and dashed curves show the output spectra with pump and probe pulses and with only a pump pulse, respectively. The pump pulsewidths are 1 ps, 0.5 ps, and 0.2 ps and the probe pulsewidth is 1 ps. The input pump and probe pulse energies are 1 pJ and 10 fJ, respectively. For a pump pulsewidth of 1 ps, the spectral peaks of the pump, probe, and FWM signals are clearly separated. The FWM signal can be obtained by the spectral filtering whose bandwidth is from +2 THz to +4 THz, which is shown in the figure by the arrow. With the decrease in the pump pulsewidth, the pump spectral width is broadened and it becomes difficult to extract the FWM signal using the optical filter due to the spectral overlap. For the shorter pump

Figure 6(b) shows the temporal waveforms of the output signals after filtering. The solid and dashed curves show the waveforms with probe and without probe pulses, respectively. The contrast between the output power with probe and without probe pulses for a pump pulsewidth of 1 ps is larger than that for the shorter pump pulsewidths. This is due to the strong overlap between pump pulse of 0.5 ps and 0.2 ps and the FWM signal in the frequency domain. For an input pump pulsewidth of 1 ps, a FWM signal pulsewidth of 0.73 ps is narrower than the input pump pulsewidth. This is due to the fact that the FWM signal

the FWM signal intensity, *pI* is the pump pulse intensity, and *qI* is the probe pulse intensity. For input pump pulsewidths of 0.5 ps and 0.2 ps, the optical bandpass filter broadens the FWM signal pulsewidth due to the limitation in the frequency domain. Then, FWM signal pulsewidths of 0.57 ps and 0.55 ps become broader than the input pump pulsewidths. By this filtering, the energies of the transform-limited sech2 pulses with pulsewidths of 1 ps, 0.73 ps, 0.5 ps, and 0.2 ps are reduced by 0.002%, 0.05%, 0.7%, and 19%, respectively. The peak powers are decreased by 0.86%, 4%, 14%, and 63%, respectively. The pulsewidth of 0.73 ps corresponds the that of the FWM pulse among 1 ps pump and 1 ps probe pulses (Fig. 6(b)). Therefore, the waveform distortion by this filtering is negligibly

Probe Pump


2 ~ 4 THz

*FWM p q I II* as reported (Das *et al.,* 2000). Here, *IFWM* is

t

FWM Signal

FWM Signal

t

t

pulsewidth less than 0.5 ps, the FWM signals are not clearly observed.

*p q I I* i.e., <sup>2</sup>

small for the FWM pulses among 1 ps pump and 1 ps probe pulses.

Probe

Pump

pulsewidth is varied.

intensity is proportional to <sup>2</sup>

Fig. 6. (a) Output spectra of the SOA before filtering. The solid and dashed curves show the output spectra with pump and probe pulses and with only a pump pulse, respectively. (b) Output pulse waveforms after filtering from +2 ~ +4 THz. The solid and dashed curves show the output pulse waveforms with pump and probe pulses and with only a pump pulse, respectively. Here, the input pump and probe pulse energies are 1 pJ and 10 fJ, respectively. The input probe pulsewidth is 1 ps.

Optical Demultiplexing Based on Four-Wave Mixing in Semiconductor Optical Amplifiers 179

efficiency increases. However, the pump pulses narrower than 0.8 ps are not suitable for a practical DEMUX operation, which is due to the low ON-OFF ratio. To improve the ON-OFF ratio, one possible method is to increase the detuning for decreasing the spectral overlap between the pump and the FWM signal. However, in our simulation using the parameters for typical SOAs working in a 0.86 m region, the increasing in the detuning is not practical because of the limited gain bandwidth of the SOAs. It is necessary to use the SOA with wider gain bandwidth for FWM among shorter pulses. One good candidate for obtaining wider gain bandwidth is to use the SOAs with staggered thickness multiple

0.1 1 10

0.5 5

Input Pump Pulsewidth (ps)

Fig. 8. (a) ON-OFF ratio and (b) FWM conversion efficiency. The input probe pulse energies are 0.01 pJ and 0.1 pJ, and pulsewidth is 1 ps. The input pump pulse energies are

(Pump, Probe) =

Input Energy

(1, 0.01) pJ (0.1, 0.1) pJ (0.1, 0.01) pJ

(1, 0.1) pJ

(1, 0.01) pJ

(1, 0.1) pJ

Input Energy (Pump, Probe) =

(0.1, 0.1) pJ

(0.1, 0.01) pJ

quantum wells (Mikami *et al.,*1991; Gingrich *et al.,* 1997).

(a)

0

0

(b)




FWM Conversion Efficiency (dB)

0.1 pJ and 1 pJ.



20

40

ON-OFF Ratio (dB)

60

80

100

Fig. 7. The energy, which is filtered from the +2 to +4 THz component of the output spectrum, against the input pump pulsewidth. The input probe pulse energies are 0.01 pJ, and 0.1 pJ, and pulsewidth is 1 ps. The input pump energies are 0.1 pJ and 1 pJ. The solid lines represent the FWM signal energy with the input probe pulse and the dashed lines represent the FWM signal energy without input probe pulse.

Figure 7 shows the energy, which is filtered from the +2 to +4 THz component of the output spectrum (as shown in Fig. 6(a)) versus the input pump pulsewidth characteristics. The solid and dashed curves show the output pulse energy with and without probe pulses, respectively. Here, the input probe pulse energies were set to be 0.01 pJ, and 0.1pJ with a pulsewidth of 1 ps and the input pump pulse energies were set to be 0.1 pJ and 1 pJ. With the decrease of pump pulsewidth, the output energy increases, while the differences between the output energy with probe pulses and without probe pulses decrease because of the overlap of the pump and the FWM signal in the spectral domain. Therefore, the regular DEMUX operation is not obtained for the pump pulsewidth of less than 0.5 ps.

Figure 8(a) shows the ON-OFF ratio and the FWM conversion efficiency characteristics. Here, the "ON-OFF ratio" is defined as the ratio of the output energy having a spectral component of +2 ~ +4 THz with the probe pulse to without the probe pulse. Therefore, in the ideal case, the output energy with the probe pulse corresponds to the FWM signal energy and the output energy without the probe pulse becomes zero. The larger ON-OFF ratio is preferable in the DEMUX operations. We have assumed that the ON-OFF ratio > 20 dB is acceptable for a practical DEMUX operation. To obtain the enough ON-OFF ratio, the pump pulsewidth should be wider than 0.8 ps for input pump pulse energies of 0.1 pJ and 1 pJ. Fig. 8(b) shows the FWM conversion efficiency increases with the increase of input pump pulse energy. The FWM signal intensity is proportional to the square of the input pump intensity (Das *et al.,* 2000). Therefore, the FWM conversion efficiency increases about 20 dB for the increase in a pump pulse energy of 10 dB in the region where the enough ON-OFF ratio is obtained. For the narrower pump pulsewidth, the nominal FWM conversion

Input Pump = 1 pJ

0.1 pJ

Input Probe =

0.1 pJ

0.01 pJ

0.1 pJ

0.01 pJ

0.1 1 10

0.5 5

Input Pump Pulsewidth (ps)

Figure 7 shows the energy, which is filtered from the +2 to +4 THz component of the output spectrum (as shown in Fig. 6(a)) versus the input pump pulsewidth characteristics. The solid and dashed curves show the output pulse energy with and without probe pulses, respectively. Here, the input probe pulse energies were set to be 0.01 pJ, and 0.1pJ with a pulsewidth of 1 ps and the input pump pulse energies were set to be 0.1 pJ and 1 pJ. With the decrease of pump pulsewidth, the output energy increases, while the differences between the output energy with probe pulses and without probe pulses decrease because of the overlap of the pump and the FWM signal in the spectral domain. Therefore, the regular

Figure 8(a) shows the ON-OFF ratio and the FWM conversion efficiency characteristics. Here, the "ON-OFF ratio" is defined as the ratio of the output energy having a spectral component of +2 ~ +4 THz with the probe pulse to without the probe pulse. Therefore, in the ideal case, the output energy with the probe pulse corresponds to the FWM signal energy and the output energy without the probe pulse becomes zero. The larger ON-OFF ratio is preferable in the DEMUX operations. We have assumed that the ON-OFF ratio > 20 dB is acceptable for a practical DEMUX operation. To obtain the enough ON-OFF ratio, the pump pulsewidth should be wider than 0.8 ps for input pump pulse energies of 0.1 pJ and 1 pJ. Fig. 8(b) shows the FWM conversion efficiency increases with the increase of input pump pulse energy. The FWM signal intensity is proportional to the square of the input pump intensity (Das *et al.,* 2000). Therefore, the FWM conversion efficiency increases about 20 dB for the increase in a pump pulse energy of 10 dB in the region where the enough ON-OFF ratio is obtained. For the narrower pump pulsewidth, the nominal FWM conversion

Fig. 7. The energy, which is filtered from the +2 to +4 THz component of the output spectrum, against the input pump pulsewidth. The input probe pulse energies are 0.01 pJ, and 0.1 pJ, and pulsewidth is 1 ps. The input pump energies are 0.1 pJ and 1 pJ. The solid lines represent the FWM signal energy with the input probe pulse and the dashed lines

DEMUX operation is not obtained for the pump pulsewidth of less than 0.5 ps.

10-1

Without Probe

represent the FWM signal energy without input probe pulse.

10<sup>0</sup>

10<sup>1</sup>

10<sup>2</sup>

Energy (aJ) [Filtered from 2 ~ 4 THz]

10<sup>3</sup>

10<sup>4</sup>

10<sup>5</sup>

efficiency increases. However, the pump pulses narrower than 0.8 ps are not suitable for a practical DEMUX operation, which is due to the low ON-OFF ratio. To improve the ON-OFF ratio, one possible method is to increase the detuning for decreasing the spectral overlap between the pump and the FWM signal. However, in our simulation using the parameters for typical SOAs working in a 0.86 m region, the increasing in the detuning is not practical because of the limited gain bandwidth of the SOAs. It is necessary to use the SOA with wider gain bandwidth for FWM among shorter pulses. One good candidate for obtaining wider gain bandwidth is to use the SOAs with staggered thickness multiple quantum wells (Mikami *et al.,*1991; Gingrich *et al.,* 1997).

Fig. 8. (a) ON-OFF ratio and (b) FWM conversion efficiency. The input probe pulse energies are 0.01 pJ and 0.1 pJ, and pulsewidth is 1 ps. The input pump pulse energies are 0.1 pJ and 1 pJ.

Optical Demultiplexing Based on Four-Wave Mixing in Semiconductor Optical Amplifiers 181

pulsewidth is due to the fact that the pulsewidth of the probe pulses (1 ps) is set to be short compared with a bit interval of 4 ps (250 Gbit/s). On the other hand, with the decreasing in the pump pulsewidth, the ON-OFF ratio severely decreases due to the overlap in the frequency domain between the pump and the FWM signal pulse as explained in Fig. 8. This small allowance is attributed to the fact that the pump pulse energy is much stronger than that of the FWM signal. These results have an interesting information; the overlap in the frequency domain is more important than the overlap in the time domain for the design of the ultrafast all-optical DEMUX. As a result of the simulation, the optimum input pump

0.1 1 10

0.5 5

Input Pump Pulsewidth (ps)

Fig. 10. ON-OFF ratio for a three-bit-stream probe. The input probe energies are 0.01 pJ and

In the experiments reported so far, 100 – 6.3 Gbit/s (Kawanishi *et al.*, 1994; Uchiyama *et al.,*  1998), 200 – 6.3 Gbit/s (Morioka *et al.,* 1996), 40 – 10 Gbit/s (Tomkos *et al.,* 1999), and 100 – 10 Gbit/s (Kirita *et al.,* 1998) demultiplexing were performed. In our simulation/ modeling, we have considered the nonlinear effects, CD, CH and SHB with the recovery times of 200 ps, 700 fs, and 60 fs, respectively (Hong *et al.,* 1996). Because, we assumed a probe pulse repetition rate of 250 Gbit/s, which is much faster than the recovery time of the CD, the CD caused by the probe pulses remains when the following probe pulses are injected into the SOA. Therefore, the pattern effect may arise and deteriorate the DEMUX operation for the multi-bit probe pulses (Saleh & Habbab, 1990). Here, we have considered the pattern effect of the probe bits because the different number of probe pulses is injected between the consecutive pump pulses depends on the bit pattern. Figure 11 shows the schematic diagram for the simulation of an optical DEMUX to investigate the pattern effect appearing at the DEMUX signals. The repetition rate and the pulsewidth of the input probe pulse are set to be 250 Gbit/s and 1 ps, respectively. We have simulated the FWM signals for the case that the different number of the probe pulses, n-1 are injected before the demultiplexed signal is extracted. More number of probe pulses reduce the DEMUX (FWM) signals as

0.1 pJ, and pulsewidth is 1 ps. The input pump pulse energies are 0.1 pJ and 1 pJ.

pulsewidth range is 1 ps ~ 3 ps for an input probe pulsewidth of 1 ps.

Input Energy

(Pump, Probe) =

(0.1, 0.01) pJ (1, 0.01) pJ (0.1, 0.1) pJ (1, 0.1) pJ

0

shown by the dashed line (where, n = 30).

10

20

30

ON-OFF Ratio (dB)

40

50

## **4. Optical DEMUX characteristics in SOAs with multi-bit probe or pump pulses**

In this section, we will discuss the optical DEMUX characteristics in SOAs with multi-bit probe or multi-bit pump pulses and vice versa. The FWM signal generates only when the pump and probe pulses are injected simultaneously into SOAs. Therefore, all-optical demultiplexed signals can be extracted as the FWM signals from a time-multiplexed signal train as described in the Introduction. Here, the pump and probe pulses act as gating and gated pulses, respectively. For a solitary probe pulse, the overlap between the pump pulse and the FWM signal in the frequency domain decreases the ON-OFF ratio as described in the previous section. The overlap in the frequency domain increases with the decrease in the pump pulsewidth. Therefore, the ON-OFF ratio increases with the increase of pump pulsewidth. On the otherhand, for the multi-bit probe pulses, the overlap among the pulses in the time domain also decreases the ON-OFF ratio. This overlap mainly comes from the neighboring pulses in the time domain. To investigate the influence of the neighboring pulses, we simulate the optical DEMUX characteristics for a three-bit-stream of 250 Gbit/s in this section. We also evaluate the pattern effect on the DEMUX (based on FWM) signals caused by the probe pulses.

Figure 9 shows the schematic diagram for the simulation of the ON-OFF ratio of the alloptical DEMUX. The probe pulses are a three-bit-stream of 250 Gbit/s. The peak position of the pump pulse is adjusted to that of the center pulse of the three probe pulses. Here, the ON-OFF ratio is defined as the ratio of the FWM signal energy obtained with the central input probe pulse of the three-bit-stream to the one obtained without the central input probe pulse. For the wider input pump pulse as indicated by the dashed lines, the FWM signal decreases and the ON-OFF ratio decreases, i.e., the crosstalk from the neighboring pulses increases.

Fig. 9. A simple schematic diagram for the simulation of ON-OFF ratio of all-optical DEMUX. The input probe pulse repetition rate is 250 Gbit/s and pulsewidth is 1 ps. In the input probe pulse stream, **'0'** represents the signal is OFF and **'1'** represents the signal is ON.

Figure 10 shows the calculated ON-OFF ratio versus the input pump pulsewidth characteristics for the three-bit-stream. The input probe pulse energies are 0.1 pJ and 0.01 pJ, and pulsewidth is 1 ps. The input pump energies are 0.01 pJ, 0.1 pJ, and 1 pJ. For the wider input pump pulsewidth, the ON-OFF ratio decreases due to the overlap in the time domain among the pump and the neighboring of probe pulses. For an input pump pulsewidth of 3 ps, the ON-OFF ratio becomes about 20 dB. This relatively large allowance in the pump

In this section, we will discuss the optical DEMUX characteristics in SOAs with multi-bit probe or multi-bit pump pulses and vice versa. The FWM signal generates only when the pump and probe pulses are injected simultaneously into SOAs. Therefore, all-optical demultiplexed signals can be extracted as the FWM signals from a time-multiplexed signal train as described in the Introduction. Here, the pump and probe pulses act as gating and gated pulses, respectively. For a solitary probe pulse, the overlap between the pump pulse and the FWM signal in the frequency domain decreases the ON-OFF ratio as described in the previous section. The overlap in the frequency domain increases with the decrease in the pump pulsewidth. Therefore, the ON-OFF ratio increases with the increase of pump pulsewidth. On the otherhand, for the multi-bit probe pulses, the overlap among the pulses in the time domain also decreases the ON-OFF ratio. This overlap mainly comes from the neighboring pulses in the time domain. To investigate the influence of the neighboring pulses, we simulate the optical DEMUX characteristics for a three-bit-stream of 250 Gbit/s in this section. We also evaluate the pattern effect on the DEMUX (based on FWM) signals

Figure 9 shows the schematic diagram for the simulation of the ON-OFF ratio of the alloptical DEMUX. The probe pulses are a three-bit-stream of 250 Gbit/s. The peak position of the pump pulse is adjusted to that of the center pulse of the three probe pulses. Here, the ON-OFF ratio is defined as the ratio of the FWM signal energy obtained with the central input probe pulse of the three-bit-stream to the one obtained without the central input probe pulse. For the wider input pump pulse as indicated by the dashed lines, the FWM signal decreases and the ON-OFF ratio decreases, i.e., the crosstalk from the neighboring pulses

t

Fig. 9. A simple schematic diagram for the simulation of ON-OFF ratio of all-optical DEMUX. The input probe pulse repetition rate is 250 Gbit/s and pulsewidth is 1 ps. In the input probe pulse stream, **'0'** represents the signal is OFF and **'1'** represents the signal is ON.

Figure 10 shows the calculated ON-OFF ratio versus the input pump pulsewidth characteristics for the three-bit-stream. The input probe pulse energies are 0.1 pJ and 0.01 pJ, and pulsewidth is 1 ps. The input pump energies are 0.01 pJ, 0.1 pJ, and 1 pJ. For the wider input pump pulsewidth, the ON-OFF ratio decreases due to the overlap in the time domain among the pump and the neighboring of probe pulses. For an input pump pulsewidth of 3 ps, the ON-OFF ratio becomes about 20 dB. This relatively large allowance in the pump

t

t

Pump

'1' '1''1'

'1' '0' '1'


**~ ~** 2 ~ 4 THz t

FWM Signal

FWM Signal

Probe Pump

SOA **~**

**4. Optical DEMUX characteristics in SOAs with multi-bit probe or pump** 

**pulses** 

caused by the probe pulses.

increases.

250 Gbit/s Probe

pulsewidth is due to the fact that the pulsewidth of the probe pulses (1 ps) is set to be short compared with a bit interval of 4 ps (250 Gbit/s). On the other hand, with the decreasing in the pump pulsewidth, the ON-OFF ratio severely decreases due to the overlap in the frequency domain between the pump and the FWM signal pulse as explained in Fig. 8. This small allowance is attributed to the fact that the pump pulse energy is much stronger than that of the FWM signal. These results have an interesting information; the overlap in the frequency domain is more important than the overlap in the time domain for the design of the ultrafast all-optical DEMUX. As a result of the simulation, the optimum input pump pulsewidth range is 1 ps ~ 3 ps for an input probe pulsewidth of 1 ps.

Fig. 10. ON-OFF ratio for a three-bit-stream probe. The input probe energies are 0.01 pJ and 0.1 pJ, and pulsewidth is 1 ps. The input pump pulse energies are 0.1 pJ and 1 pJ.

In the experiments reported so far, 100 – 6.3 Gbit/s (Kawanishi *et al.*, 1994; Uchiyama *et al.,*  1998), 200 – 6.3 Gbit/s (Morioka *et al.,* 1996), 40 – 10 Gbit/s (Tomkos *et al.,* 1999), and 100 – 10 Gbit/s (Kirita *et al.,* 1998) demultiplexing were performed. In our simulation/ modeling, we have considered the nonlinear effects, CD, CH and SHB with the recovery times of 200 ps, 700 fs, and 60 fs, respectively (Hong *et al.,* 1996). Because, we assumed a probe pulse repetition rate of 250 Gbit/s, which is much faster than the recovery time of the CD, the CD caused by the probe pulses remains when the following probe pulses are injected into the SOA. Therefore, the pattern effect may arise and deteriorate the DEMUX operation for the multi-bit probe pulses (Saleh & Habbab, 1990). Here, we have considered the pattern effect of the probe bits because the different number of probe pulses is injected between the consecutive pump pulses depends on the bit pattern. Figure 11 shows the schematic diagram for the simulation of an optical DEMUX to investigate the pattern effect appearing at the DEMUX signals. The repetition rate and the pulsewidth of the input probe pulse are set to be 250 Gbit/s and 1 ps, respectively. We have simulated the FWM signals for the case that the different number of the probe pulses, n-1 are injected before the demultiplexed signal is extracted. More number of probe pulses reduce the DEMUX (FWM) signals as shown by the dashed line (where, n = 30).

Optical Demultiplexing Based on Four-Wave Mixing in Semiconductor Optical Amplifiers 183

carrier density at *t = 0* (before the injection of probe pulse), and *a* is the coefficient of the

<sup>0</sup> 1 exp *<sup>N</sup> aP <sup>N</sup> aP t*

The repetition rate of the probe pulse is 250 GHz. Thus, we assume that the light with a constant photon density *P* is injected. The duration of the probe bits is given by <sup>11</sup> *t n* (2.5 10 ) . Because, the FWM signal intensity is proportional to the carrier density *N*,

*S tA B t FWM*( ) 1 exp /

Here, *A(1+B)* corresponds to the maximum FWM signal intensity at *t = 0*, *B* is the constant representing the decrease in the FWM signal intensity caused by the probe pulses and

the effective recovery time of the carrier density depending on the input probe intensity. From equation (20), we obtained that the maximum fluctuation reaches to ~15% for the

(0) ( )

*S S B*

Figure 13 shows another example of the pattern effects on the DEMUX signal for an input pump energy of 1 pJ and a probe energy of 0.01 pJ. In this case, the FWM signal intensities are stronger than the results shown in Fig. 12, because the probe energy is 10 times lower and the pump energy is 10 times as stronger than in Fig. 12. The FWM signal energy decreases by only 0.03% for 30 probe pulses. We have obtained FWM signal energy *A* of 62.0

energy reduces only by 1.14% for the infinite number of probe pulses. We believe that such a small fluctuation is not an obstacle for the practical application. Although the results are not shown here, another set of the calculations are carried out shown in the Fig. 13, where the input pump and probe energy are 1 pJ and 0.1 pJ, respectively. The input pump energy is 10 times stronger than that of Fig. 12. The FWM signal intensities were about 100 times stronger than the results shown in Fig. 12. The FWM signal peak power decreases by less than 3% for 30 probe pulses. From the fitting to the calculations, we have obtained that *A* is

infinite number of probe pulses in this condition. From these results, we can conclude that the intensity fluctuation of the FWM signal can be decreased by using the strong pump

*FWM FWM FWM*

(0) 1

of 200 ps from the fitted curve of Fig. 13(b). In this case, the FWM signal

is 175 ps. Therefore, the FWM signal energy reduces by 9% for the

is defined as

is 200 ps and agrees with

1 *aP*

should correspond to the carrier recovery time

. Therefore, in

s. In the case of the

s. In

*S B*

curve using the equation (19) with the parameters, *A* of 18.0 aJ, *B* of 0.178, and

 

stimulated emission. At *t = 0*, *N* = *N0*, then the solution of equation (17) is as follows:

s) is the recovery rate of the carrier density, *N0* is the

(19)

. In Fig. 12(b) the solid line shows the fitted

(20)

(18)

of 172 ps.

is

 (= 1/

> *aP*

the FWM signal intensity *SFWM* can be expressed as follows.

 and 1 *aP*

where, *P* is the photon density,

where, *A N*

aJ, *B* of 0.012, and

0.565 fJ, *B* is 0.09, and

 <sup>0</sup> , *B aP*

infinite number of probe pulse train.

pulses or/and the weak input probe pulses.

the weak probe case as shown in Fig. 13(b),

the weak limit of the probe pulses,

The effective recovery time of the carrier density

Fig. 11. Schematic diagram for the simulation of pattern effect on all-optical DEMUX operation. The input probe pulse repetition rate and pulsewidth are 250 Gbit/s and 1 ps, respectively. The different number of the probe pulses n-1 are injected before the demultiplexed signal is extracted.

Figure 12(a) shows an example of the pattern effect on the DEMUX signal waveforms. The input pump and probe pulse energies are 0.1 pJ (Ep) and 0.1 pJ (Eq), respectively. With increase the number of probe pulses, the FWM signal peak power decreases. The reduction in the peak power amounts to 7.4% for 30 probe pulses, while the waveforms remain unaffected. Fig. 12(b) shows the FWM signal energy versus the number of probe pulses. The closed circles show the calculated results and the solid line shows the fitted curve under the following approximation (Das, 2000; Das *et al.,* 2001). The FWM signal is generated through the modulations in the refractive index and gain in the active region of SOAs. The modulation depths are proportional to both the carrier density and photon density because the modulation is created by the stimulated emission. Therefore, the FWM signal may also be proportional to the carrier density and photon density. The rate equation that describes the carrier density *N* in the active region is given by

$$\frac{dN}{dt} = \beta(N\_0 - N) - aPN \tag{17}$$

Fig. 12. (a) Pattern effect on the DEMUX signal waveforms and (b) FWM signal energy against number probe pulses. The input pump and probe energies are 0.1 pJ and 0.1 pJ, respectively.

t

Fig. 11. Schematic diagram for the simulation of pattern effect on all-optical DEMUX operation. The input probe pulse repetition rate and pulsewidth are 250 Gbit/s and 1 ps,

Figure 12(a) shows an example of the pattern effect on the DEMUX signal waveforms. The input pump and probe pulse energies are 0.1 pJ (Ep) and 0.1 pJ (Eq), respectively. With increase the number of probe pulses, the FWM signal peak power decreases. The reduction in the peak power amounts to 7.4% for 30 probe pulses, while the waveforms remain unaffected. Fig. 12(b) shows the FWM signal energy versus the number of probe pulses. The closed circles show the calculated results and the solid line shows the fitted curve under the following approximation (Das, 2000; Das *et al.,* 2001). The FWM signal is generated through the modulations in the refractive index and gain in the active region of SOAs. The modulation depths are proportional to both the carrier density and photon density because the modulation is created by the stimulated emission. Therefore, the FWM signal may also be proportional to the carrier density and photon density. The rate equation that describes

> <sup>0</sup> ( ) *dN N N aPN dt*

> > 0

10

FWM Signal Energy (aJ)

20

30

(b)

1 30

Fig. 12. (a) Pattern effect on the DEMUX signal waveforms and (b) FWM signal energy against number probe pulses. The input pump and probe energies are 0.1 pJ and 0.1 pJ,

respectively. The different number of the probe pulses n-1 are injected before the

Pump

demultiplexed signal is extracted.

0

respectively.

10

20

FWM Signal Power (

W)

30

Ep = 0.1 pJ Eq = 0.1 pJ

40

the carrier density *N* in the active region is given by

(a) No. of Probe Pulses


Time (ps)

250 Gbit/s Probe

1 n-2 n-1 n

.....

t


SOA **~** 

Probe FWM

Pump

Signal

**~ ~** 

2 ~ 4 THz

f (THz)

t

n = 1 n = 30

DEMUX Signal

(17)

Calculated Results Fitted Curve

1 10 100

Number of Probe Pulses

where, *P* is the photon density, (= 1/s) is the recovery rate of the carrier density, *N0* is the carrier density at *t = 0* (before the injection of probe pulse), and *a* is the coefficient of the stimulated emission. At *t = 0*, *N* = *N0*, then the solution of equation (17) is as follows:

$$N = \frac{\beta N\_0}{aP + \beta} \left[ 1 + \frac{aP}{\beta} \exp\left\{-\left(aP + \beta\right)t\right\} \right] \tag{18}$$

The repetition rate of the probe pulse is 250 GHz. Thus, we assume that the light with a constant photon density *P* is injected. The duration of the probe bits is given by <sup>11</sup> *t n* (2.5 10 ) . Because, the FWM signal intensity is proportional to the carrier density *N*, the FWM signal intensity *SFWM* can be expressed as follows.

$$S\_{\rm FWHM}(t) = A \boxed{1 + B \exp\left(-t \;/\; \tau\right)} \tag{19}$$

where, *A N* <sup>0</sup> , *B aP* and 1 *aP* . In Fig. 12(b) the solid line shows the fitted curve using the equation (19) with the parameters, *A* of 18.0 aJ, *B* of 0.178, and of 172 ps. Here, *A(1+B)* corresponds to the maximum FWM signal intensity at *t = 0*, *B* is the constant representing the decrease in the FWM signal intensity caused by the probe pulses and is the effective recovery time of the carrier density depending on the input probe intensity. From equation (20), we obtained that the maximum fluctuation reaches to ~15% for the infinite number of probe pulse train.

$$\frac{S\_{\text{FWHM}}(\mathbf{0}) - S\_{\text{FWHM}}(\infty)}{S\_{\text{FWHM}}(\mathbf{0})} = \frac{B}{1 + B} \tag{20}$$

Figure 13 shows another example of the pattern effects on the DEMUX signal for an input pump energy of 1 pJ and a probe energy of 0.01 pJ. In this case, the FWM signal intensities are stronger than the results shown in Fig. 12, because the probe energy is 10 times lower and the pump energy is 10 times as stronger than in Fig. 12. The FWM signal energy decreases by only 0.03% for 30 probe pulses. We have obtained FWM signal energy *A* of 62.0 aJ, *B* of 0.012, and of 200 ps from the fitted curve of Fig. 13(b). In this case, the FWM signal energy reduces only by 1.14% for the infinite number of probe pulses. We believe that such a small fluctuation is not an obstacle for the practical application. Although the results are not shown here, another set of the calculations are carried out shown in the Fig. 13, where the input pump and probe energy are 1 pJ and 0.1 pJ, respectively. The input pump energy is 10 times stronger than that of Fig. 12. The FWM signal intensities were about 100 times stronger than the results shown in Fig. 12. The FWM signal peak power decreases by less than 3% for 30 probe pulses. From the fitting to the calculations, we have obtained that *A* is 0.565 fJ, *B* is 0.09, and is 175 ps. Therefore, the FWM signal energy reduces by 9% for the infinite number of probe pulses in this condition. From these results, we can conclude that the intensity fluctuation of the FWM signal can be decreased by using the strong pump pulses or/and the weak input probe pulses.

The effective recovery time of the carrier density is defined as 1 *aP* . Therefore, in the weak limit of the probe pulses, should correspond to the carrier recovery time s. In the weak probe case as shown in Fig. 13(b), is 200 ps and agrees with s. In the case of the

Optical Demultiplexing Based on Four-Wave Mixing in Semiconductor Optical Amplifiers 185

level on the assumption that the ASE level is – 40 dBm/nm (Diez *et al.,* 1997). In Fig. 6(a), the FWM signal is directly compared with ASE level. The FWM signal is about 10 dB greater than the ASE level when the pump pulsewidth is 1 ps. In Fig. 6(b), the ASE level becomes ~ 0.5 W if we use a filter with 2 THz bandwidth. We can observe very clearly the FWM signal in the time domain. In Fig. 7, the ASE level becomes ~ 8 aJ if we use a filtering with 2 THz bandwidth and select a time slot of 16 ps (i.e., consider a 62.5 GHz repetition rate). Therefore, except for the conditions of a 0.1 pJ pump pulse and a 0.01 pJ probe pulse, the energy of FWM signal is greater than the ASE level. For more detailed comparison, it is recommended to take into account the ASE effect and its dynamic characteristics in the

simulation.

0

1

0

100

0

0 50 100 150 200 250

Time (ps)

Fig. 14. DEMUX signal characteristics for the repetitive pump pulses. The input pump and probe pulse energies are 1 pJ and 0.01 pJ, respectively. Here, top figure is the input probe pulses, middle figure is the input pump pulses and bottom figure is the generated FWM

20

40

FWM Signal Pow

signal pulses.

 er (

W)

60

80

0.2

0.4

Input Pump Pow

 er (W)

0.6

0.8

2

4

Input Probe Pow

6

 er (mW)

8

10

strong probe pulses, becomes short due to the stimulated emission caused by the probe pulses. For a strong probe pulse energy of 0.1 pJ, becomes smaller and they are 172 ps and 175 ps for pump pulse energies of 1 pJ and 0.1 pJ, respectively. These results support our assumptions as mentioned above.

Fig. 13. (a) Pattern effect on the DEMUX signal waveforms and (b) FWM signal energy against number probe pulses. For this case, the input pump and probe energies are 1 pJ and 0.01 pJ, respectively.

Figure 14 shows an example of temporal waveforms of demuliplexed signals from time multiplexed signals by repetitive pump pulses. The input pump and probe pulse energies are 1 pJ and 0.01 pJ, respectively. The probe pulses are with a pulsewidth of 1 ps, sech2 shape and have a repetition rate of 250 GHz. The pump pulses are with a pulsewidth of 1 ps, sech2 shape and have a repetition rate of 62.5 GHz. Therefore, the 62.5 Gbit/s demultiplexed signals are selected once every four bits from the 250 Gbit/s signals. The FWM signal power is decreased by the strong input pump power due to the gain saturation and reaches to the constant value which is ~23% of the FWM signal power among the solitary pulses. There will be no pattern effect due to the gain saturation caused by the pump power, because the pump pulses are injected continuously. In this particular case with a low probe pulse energy of 0.01 pJ, the pattern effect caused by the probe pulse is expected to be very small as shown in Fig. 13.

One of the most important effects we have not included in this modeling is an amplified spontaneous emission (ASE) noise which is generated in SOAs. However, a number of literature emphasized the importance of the ASE noise. Summerfield and Tucker (Summerfield & Tucker, 1995) defined and measured the noise figure of an optical frequency converter based on FWM in an SOA. Diez *et al.,* (Diez *et al.,* 1997) defined the signal-to-background ratio (SBR) and investigated that different optimization criteria than for continuous waves apply as far as pulsed FWM applications concerned. Diez et al., (Diez *et al.,* 1999) also reported a strong dependence of both conversion efficiency and SBR on pulsewidth and bit rate. This behavior has been attributed to the dynamics of the ASE, which is the main source of noise in an SOA.

Although the level of ASE strongly depends on the SOA structure and the operation conditions of SOAs, we have roughly compared with our calculated results and the ASE

175 ps for pump pulse energies of 1 pJ and 0.1 pJ, respectively. These results support our

1 30

Fig. 13. (a) Pattern effect on the DEMUX signal waveforms and (b) FWM signal energy against number probe pulses. For this case, the input pump and probe energies are 1 pJ and

Figure 14 shows an example of temporal waveforms of demuliplexed signals from time multiplexed signals by repetitive pump pulses. The input pump and probe pulse energies are 1 pJ and 0.01 pJ, respectively. The probe pulses are with a pulsewidth of 1 ps, sech2 shape and have a repetition rate of 250 GHz. The pump pulses are with a pulsewidth of 1 ps, sech2 shape and have a repetition rate of 62.5 GHz. Therefore, the 62.5 Gbit/s demultiplexed signals are selected once every four bits from the 250 Gbit/s signals. The FWM signal power is decreased by the strong input pump power due to the gain saturation and reaches to the constant value which is ~23% of the FWM signal power among the solitary pulses. There will be no pattern effect due to the gain saturation caused by the pump power, because the pump pulses are injected continuously. In this particular case with a low probe pulse energy of 0.01 pJ, the pattern effect caused by the probe pulse is expected to be very small as shown in Fig. 13.

One of the most important effects we have not included in this modeling is an amplified spontaneous emission (ASE) noise which is generated in SOAs. However, a number of literature emphasized the importance of the ASE noise. Summerfield and Tucker (Summerfield & Tucker, 1995) defined and measured the noise figure of an optical frequency converter based on FWM in an SOA. Diez *et al.,* (Diez *et al.,* 1997) defined the signal-to-background ratio (SBR) and investigated that different optimization criteria than for continuous waves apply as far as pulsed FWM applications concerned. Diez et al., (Diez *et al.,* 1999) also reported a strong dependence of both conversion efficiency and SBR on pulsewidth and bit rate. This behavior has been attributed to the dynamics of the ASE,

Although the level of ASE strongly depends on the SOA structure and the operation conditions of SOAs, we have roughly compared with our calculated results and the ASE

No. of Probe Pulses

0

20

40

FWM Signal Energy (aJ)

60

80

100

becomes short due to the stimulated emission caused by the probe

(b)

becomes smaller and they are 172 ps and

Calculated Results Fitted Curve

1 10 100

Number of Probe Pulses

strong probe pulses,

(a) Ep = 1 pJ Eq = 0.01 pJ

0

0.01 pJ, respectively.

20

40

FWM Signal Power (

W)

60

80

100

assumptions as mentioned above.

pulses. For a strong probe pulse energy of 0.1 pJ,


Time (ps)

which is the main source of noise in an SOA.

level on the assumption that the ASE level is – 40 dBm/nm (Diez *et al.,* 1997). In Fig. 6(a), the FWM signal is directly compared with ASE level. The FWM signal is about 10 dB greater than the ASE level when the pump pulsewidth is 1 ps. In Fig. 6(b), the ASE level becomes ~ 0.5 W if we use a filter with 2 THz bandwidth. We can observe very clearly the FWM signal in the time domain. In Fig. 7, the ASE level becomes ~ 8 aJ if we use a filtering with 2 THz bandwidth and select a time slot of 16 ps (i.e., consider a 62.5 GHz repetition rate). Therefore, except for the conditions of a 0.1 pJ pump pulse and a 0.01 pJ probe pulse, the energy of FWM signal is greater than the ASE level. For more detailed comparison, it is recommended to take into account the ASE effect and its dynamic characteristics in the simulation.

Fig. 14. DEMUX signal characteristics for the repetitive pump pulses. The input pump and probe pulse energies are 1 pJ and 0.01 pJ, respectively. Here, top figure is the input probe pulses, middle figure is the input pump pulses and bottom figure is the generated FWM signal pulses.

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