**4.2. Optical phase-conjugate characteristics for linearly-chirped input probe pulse**

In this sub-section, we have analyzed the optical phase-conjugate characteristics of the FWM signal pulses. The outline of our simulation model, which is similar to the experimental setup used by Kikuchi & Matsumura (Kikuchi & Matsumura, 1998), is the following one, as shown in Fig. 7, the Fourier transform-limited optical pulse is linearly-chirped by transmission through a fiber (Fiber I) and then injected into the SOA as the probe pulse together with a pump pulse. The FWM signal is generated by the mixing of the probe pulse and the pump pulse, and selected by an optical narrow band-pass filter. The FWM signal is then transmitted through another fiber (Fiber II) that has the same GVD as Fiber I and is an appropriate length.

98 Optical Communication

pulse is shifted by about –35 GHz.

shifted toward the higher frequency side. Therefore, the center frequency of the FWM signal

Pump Probe Signal

Signal

Pump

Probe

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


Time (ps)

**4.2. Optical phase-conjugate characteristics for linearly-chirped input probe** 

In this sub-section, we have analyzed the optical phase-conjugate characteristics of the FWM signal pulses. The outline of our simulation model, which is similar to the experimental setup used by Kikuchi & Matsumura (Kikuchi & Matsumura, 1998), is the following one, as shown in Fig. 7, the Fourier transform-limited optical pulse is linearly-chirped by transmission through a fiber (Fiber I) and then injected into the SOA as the probe pulse

and probe energy levels are 1 pJ and 0.1 pJ, respectively.



(b) Phase



Frequency Shift (GHz)

0

20

40

0

0.2

0.4

0.6

Normalized Output Power

0.8

1

(a) Waveforms

**pulse** 

**Figure 7.** A simple schematic diagram for the simulation of optical phase-conjugate characteristics of picosecond FWM signal pulses in SOAs. Fourier transform-limited Gaussian optical pulse is linearlychirped by transmitting through a fiber (Fiber I) and injected into the SOA as the probe pulse together with a pump pulse. The FWM signal is generated by the mixing of the chirped-probe pulse and the pump pulse, and it is selected by an optical narrow band-pass filter. Then, the FWM signal is transmitted through the fiber (Fiber II) with the same GVD value as Fiber I and the appropriate fiber length.

We have assumed that the input probe pulse is the Fourier transform-limited Gaussian pulse with a pulsewidth of 1 ps (FWHM). Therefore, the incident field is given by Agrawal (Agrawal, 1995; Das *et al.,* 2001)

$$E(0,T) = \exp\left[-\frac{T^2}{2T\_{\text{o}}^2}\right] \tag{17}$$

where, 0 *T* is the half-width at the 1 *e* -intensity (power) point and 0.60056 ps in this case. The input probe pulse energy is chosen to be 0.1 pJ. This unchirped Gaussian pulse has propagated through the fiber (length *Zf* ) from position A to position B in Fig. 7 and is chirped by the second-order GVD of the fiber, and the duration of the pulse is broadened. The spectrum of the broadened pulse at position B is calculated using the following equation (Agrawal, 1995; Das *et al.,* 2001)

$$\tilde{E}(Z\_{f'}, \alpha) = \tilde{E}(0, \alpha) \exp\left[\frac{i}{2} \beta\_{2f} \alpha^2 Z\_{f}\right] \tag{18}$$

where, *E*(0, ) ω is the Fourier transform of the incident field given by equation (17) at *Zf* = 0 (at position A of Fig. 7), 2( ) *<sup>q</sup>* ω π = −*f f* , *<sup>q</sup> f* is the center frequency of the probe pulse, and 2 *f* β is the second-order GVD of the fiber. Equation (18) shows that GVD changes the phase of each spectral component of the pulse by an amount that depends on the frequency and the propagation distance. Even though such phase changes do not affect the pulse spectrum, they can modify the pulse shape. Taking the inverse Fourier transform of the Eq. (18), the broadened pulse can be calculated at position B of Fig. 7 using the following equation (Agrawal, 1995; Das *et al.,* 2001):

$$E(Z\_{f'},T) = \frac{1}{2\pi} \int\_{-\infty}^{\infty} \tilde{E}(0,\omega) \exp\left[\frac{i}{2} \mathcal{J}\_{2f} \phi^2 Z\_f - i\phi T\right] d\phi$$

$$= \frac{T\_0}{\left(T\_0^2 - i\mathcal{J}\_{2f} Z\_f\right)^{3/2}} \exp\left[-\frac{T^2}{\left(T\_0^2 - i\mathcal{J}\_{2f} Z\_f\right)}\right] \tag{19}$$

A Gaussian pulse preserves its shape on propagation, yet its width increases and becomes (Agrawal, 1995; Das *et al.,* 2001):

$$T\_b = T\_0 \left[ 1 + \left( Z\_f \Big/ L\_D \right)^2 \right]^{\frac{1}{2}} \tag{20}$$

where the dispersion length <sup>2</sup> *D f* 0 2 *L T*= β . The input probe pulse is chirped by the Fiber I with 2 2 2 β β*ff fD Z L* = = –0.72135 ps2 and is broadened to approximately 2.2 ps.

The FWM signal generated by the chirped probe pulse is expressed by Eq. (18) and the Fourier transform-limited Gaussian pump pulses, is obtained from the output facet of a 350 μm length SOA. The detuning between the input pump and the probe pulses is set at 3 THz. The FWM signal is obtained by taking the spectral component between +2 THz and +4 THz (i.e., the bandwidth of the optical filter is 2 THz).

Figure 8 shows the waveforms and the frequency shift of the FWM signal at the output end of the SOA, shown as position C in Fig. 7 together with those of the input probe pulse. The input probe pulse is a chirped Gaussian pulse with a pulsewidth of 2.2 ps (FWHM), as described above, and with an energy of 0.1 pJ. The input pump pulses are the Fourier transform-limited Gaussian pulses with pulsewidths of 1 ps, 2 ps, 3 ps, and 10 ps. The pulsewidth of the FWM signal is increased in step with the increase in the pump pulsewidth. The peak positions of the FWM signals are slightly shifted toward the leading edge due to the gain spectrum of the SOA. As shown in Fig. 8(b) later, the frequency of the probe pulse is linearly chirped from higher frequencies to lower frequencies. As the probe pulse frequency is set to the low frequency side of the SOA gain spectrum, the probe pulse has a larger gain at its leading edge. The center frequency shift of the FWM signal pulses at the output end of the SOA for different input pump pulsewidths is shown in Fig. 8(b), demonstrating the frequency shift of the chirped probe pulse at the input side of the SOA.

where, *E*(0, )

2 *f* β

ω

(at position A of Fig. 7), 2( ) *<sup>q</sup>*

(Agrawal, 1995; Das *et al.,* 2001):

(Agrawal, 1995; Das *et al.,* 2001):

where the dispersion length <sup>2</sup>

(i.e., the bandwidth of the optical filter is 2 THz).

 β

with 2 2 2 β

ω π= −*f f* , *<sup>q</sup>*

is the Fourier transform of the incident field given by equation (17) at *Zf* = 0

2 2

2

β

1 2 2

 ωω*Z iTd*

<sup>0</sup> <sup>1</sup> *b f <sup>D</sup> T T ZL* = + (20)

. The input probe pulse is chirped by the Fiber I

(19)

β ω

= −

( ) ( )

A Gaussian pulse preserves its shape on propagation, yet its width increases and becomes

*ff fD Z L* = = –0.72135 ps2 and is broadened to approximately 2.2 ps.

The FWM signal generated by the chirped probe pulse is expressed by Eq. (18) and the Fourier transform-limited Gaussian pump pulses, is obtained from the output facet of a 350 μm length SOA. The detuning between the input pump and the probe pulses is set at 3 THz. The FWM signal is obtained by taking the spectral component between +2 THz and +4 THz

Figure 8 shows the waveforms and the frequency shift of the FWM signal at the output end of the SOA, shown as position C in Fig. 7 together with those of the input probe pulse. The input probe pulse is a chirped Gaussian pulse with a pulsewidth of 2.2 ps (FWHM), as described above, and with an energy of 0.1 pJ. The input pump pulses are the Fourier transform-limited Gaussian pulses with pulsewidths of 1 ps, 2 ps, 3 ps, and 10 ps. The pulsewidth of the FWM signal is increased in step with the increase in the pump pulsewidth. The peak positions of the FWM signals are slightly shifted toward the leading edge due to the gain spectrum of the SOA. As shown in Fig. 8(b) later, the frequency of the probe pulse is linearly chirped from higher frequencies to lower frequencies. As the probe pulse frequency is set to the low frequency side of the SOA gain spectrum, the probe pulse has a larger gain at its leading edge. The center frequency shift of the FWM signal pulses at the output end of the SOA for different input pump pulsewidths is shown in Fig. 8(b), demonstrating the frequency shift of the chirped probe pulse at the input side of the SOA.

*T T TiZ TiZ*

= − <sup>−</sup> <sup>−</sup>

1 2 <sup>2</sup> <sup>2</sup> 0 2 0 2 exp

*f f f f*

( )

 is the second-order GVD of the fiber. Equation (18) shows that GVD changes the phase of each spectral component of the pulse by an amount that depends on the frequency and the propagation distance. Even though such phase changes do not affect the pulse spectrum, they can modify the pulse shape. Taking the inverse Fourier transform of the Eq. (18), the broadened pulse can be calculated at position B of Fig. 7 using the following equation

<sup>1</sup> ( , ) (0, )exp 2 2 *f f <sup>f</sup>*

ω

*<sup>i</sup> EZ T E*

−∞

0

β

*D f* 0 2 *L T*= β

∞

π

*f* is the center frequency of the probe pulse, and

**Figure 8.** Waveforms and frequency shift of the generated FWM signal at the output end of the SOA shown as the position C in Fig. 7 together with those of the input probe pulse.

We have defined zero frequency shift to be that at 0 ps. If the SOA acts as an ideal optical phase conjugator, the frequency shift of the FWM signal should be symmetrical with that of the input probe pulse. The frequency shift of the FWM signal is plotted for output power, which is greater than 1% of the FWM peak power. For the input pump pulsewidth of 10 ps, the frequency shift of the FWM signal is very similar to the symmetrical shape of the input probe. With a decrease in the input pump pulsewidth, the symmetry breaks due to a change in the refractive index of the SOA, as caused by the pump pulse. For a 1 ps pump pulsewidth, the symmetry is strongly degraded. In the model, we have taken into account carrier depletion, CH, SHB, and the instantaneous nonlinear Kerr effect, as the origins of nonlinear refractive index changes. For the case of a 1 ps pump pulse, the anomalous frequency shift at the leading edge of the pump pulse primarily originates from the Kerr effect. By contrast, all mechanisms contribute to the frequency shift at the trailing edge. As a result of this simulation, we found that the phase-conjugate characteristics are almost entirely preserved, even for a 2 ps input pump pulsewidth.

**Figure 9.** Spectra and phase in the frequency domain of the FWM signal at the output end of the SOA shown as the position C in Fig. 7 together with those of the probe input pulse.

The spectra and phase of the generated FWM signal in the frequency domain at the output end of the SOA shown as the position C in Fig. 7, together with the spectra and phase of the input probe pulse, as shown in Fig. 9. Although the center frequency of the probe pulse is at –3 THz, the spectrum and the phase of the input probe pulse are plotted at +3 THz to aid for comparison with the FWM signal pulse. We assumed a Fourier transform-limited Gaussian pulse as the input of the fiber, and so the power spectrum does not change during the propagation in the fiber. Therefore, the input probe spectrum is the same as the input pulse to the fiber. The phase of the probe pulse in the frequency domain should vary according to ( )<sup>2</sup> *<sup>q</sup> f* − *f* against the frequency *f* . Here, *<sup>q</sup> f* is the center frequency of the probe pulse. The results shown in Fig. 9 are in complete agreement with the above considerations. The peak frequencies of the FWM spectra are shifted to the lower frequency side of the frequency spectra. These shifts are mainly due to the SPM caused by the gain saturation effect (Das et al., 2000; Das et al., 2001). The phase of the FWM spectra at the output of the SOA is shown in Fig. 9(b). If the SOA acts as an ideal optical phase conjugator, the phase of the FWM spectra should be symmetrical with that of the input probe pulse spectrum. From the figure, the phase of the FWM signal for a pump pulsewidth of 10 ps is almost symmetrical to the phase of the input probe pulse. With a decrease in the input pump pulsewidth, this symmetry decreases. This tendency is the same as that found for time domain (see Fig. 9(a)).

102 Optical Communication

**Figure 9.** Spectra and phase in the frequency domain of the FWM signal at the output end of the SOA

Input Probe

2 2.5 3 3.5 4

Frequency (THz)

1 ps 2 ps 3 ps 10 ps

(a) Spectra at C Input Pump :

The spectra and phase of the generated FWM signal in the frequency domain at the output end of the SOA shown as the position C in Fig. 7, together with the spectra and phase of the input probe pulse, as shown in Fig. 9. Although the center frequency of the probe pulse is at –3 THz, the spectrum and the phase of the input probe pulse are plotted at +3 THz to aid for comparison with the FWM signal pulse. We assumed a Fourier transform-limited Gaussian pulse as the input of the fiber, and so the power spectrum does not change during the propagation in the fiber. Therefore, the input probe spectrum is the same as the input pulse to the fiber. The phase of the probe pulse in the frequency

frequency of the probe pulse. The results shown in Fig. 9 are in complete agreement with the above considerations. The peak frequencies of the FWM spectra are shifted to the lower frequency side of the frequency spectra. These shifts are mainly due to the SPM caused by the gain saturation effect (Das et al., 2000; Das et al., 2001). The phase of the

*<sup>q</sup> f* − *f* against the frequency *f* . Here, *<sup>q</sup>*

1 ps 2 ps 3 ps 10 ps

Input Probe

*f* is the center

shown as the position C in Fig. 7 together with those of the probe input pulse.

(b) Phase

Input Pump :

domain should vary according to ( )<sup>2</sup>

0

10



0

Phase (Radian)

5

0.2

0.4

Normalized FWM Power

0.6

0.8

1

**Figure 10.** FWM signal pulsewidth versus the β<sup>2</sup> *f f Z* value characteristics at position D in Fig. 7.

The chirped pulse can be easily compressed using the phase-conjugate characteristics of the FWM signal pulse and the second-order GVD of a fiber. Figure 10 shows the pulsewidth of the FWM signal versus the β <sup>2</sup> *f f Z* value characteristics at position D in Fig. 7. For a 10 ps pump pulse, the SOA acts as a nearly ideal phase conjugator within the confines of reversing the chirp of optical pulses. The pulsewidth of the output pulse becomes the shortest, 1.03 ps, at β <sup>2</sup> *f f Z* = –0.70 ps2 (Dijaili et al., 1990; Kikuchi & Matsumura, 1998), which is slightly smaller (~3%) than the assumed β <sup>2</sup> *f f Z* of the input fiber we assumed. The shortest pulsewidth compresses to 1 ps, which is equal to the input pulsewidth of the Fourier transform-limited probe pulse. This result confirms that the SOA acts as a nearly ideal phase conjugator (i.e., reverse chirp) when the input pump pulsewidth is relatively long (10 ps). When the input pump pulsewidth becomes shorter, the β <sup>2</sup> *f f Z* value for obtaining the shortest pulsewidth becomes smaller, and the shortest pulsewidth becomes wider than the input pulsewidth of 1 ps. For example, for a 3 ps pump pulse, the shortest pulsewidth is ~1.1 ps which is obtained at β <sup>2</sup> *f f Z* = –0.45 ps2. This result can be understood as follows: The FWM process acts as both a temporal and a spectral window depending on the pulsewidth and the spectral width of the pulses. By the FWM characteristics described in III-A, the pulsewidth of the FWM signal becomes shorter than the chirped probe pulse. Therefore, only a part of the phase information is copied to the FWM signal. For a 1 ps pump pulse, the temporal window effect is enhanced. In addition, the pump pulse loses phase information due to the optical nonlinear effect that is induced by their strong pulse peak intensity, as shown in Fig. 7. Therefore, the FWM signal pulsewidth becomes less than 1 ps at = 0 and the shortest FWM signal pulsewidth was obtained for β<sup>2</sup> *f f Z* = –0.07 ps2.

**Figure 11.** Normalized FWM signal waveforms having minimum pulsewidth (a) and the frequency shift (b) at position D in Fig. 7 after the pulse compression by the fiber. This figure also shows those of the input pulse at position A in Fig. 7, which correspond to the case that the SOA acts as the ideal phase conjugation.

Figure 11 shows the normalized FWM signal waveforms with minimum pulsewidth (a), and the frequency shift (b) at the position D in Fig. 7 after the pulse compression by the same fiber. The normalized waveform of the input pulse at position A in Fig. 7 is also shown, corresponds to the case where the SOA acts as an ideal phase conjugator. The normalized waveforms of the FWM signal pulse for various pump pulsewidths are shown in the figure. Although all the FWM signal pulses are compressed to ~ 1 ps when the values of β <sup>2</sup> *f f Z* is optimized, the FWM signal waveform is close to that of the ideal phase conjugator for the longer pump pulse of 10 ps. The frequency shift of the FWM signals after fiber dispersion compensation shows that for the longer pump pulse of 10 ps, the frequency shift becomes nearly zero, which indicates that chirp becomes nearly zero. This suggests that the compressed FWM signal almost becomes a transform-limited pulse. For the pump pulsewidth of 1 ps, the phase of the FWM signal was significantly distorted.

104 Optical Communication

conjugation.

pulsewidth is ~1.1 ps which is obtained at

0




Frequency Shift (GHz)

0

100

200

300

0.2

0.4

Normalized FWM Power

0.6

0.8

1

wider than the input pulsewidth of 1 ps. For example, for a 3 ps pump pulse, the shortest

as follows: The FWM process acts as both a temporal and a spectral window depending on the pulsewidth and the spectral width of the pulses. By the FWM characteristics described in III-A, the pulsewidth of the FWM signal becomes shorter than the chirped probe pulse. Therefore, only a part of the phase information is copied to the FWM signal. For a 1 ps pump pulse, the temporal window effect is enhanced. In addition, the pump pulse loses phase information due to the optical nonlinear effect that is induced by their strong pulse peak intensity, as shown in Fig. 7. Therefore, the FWM signal pulsewidth becomes less than

(a) Waveforms at D Input Pump :

1 ps

Input Pump : (b) Phase

<sup>2</sup> *f f Z* = –0.45 ps2. This result can be understood

β

1 ps 2 ps 3 ps 10 ps

Ideal Case

1 ps 2 ps 3 ps 10 ps

Ideal Case

<sup>2</sup> *f f Z* = –0.07 ps2.

β

**Figure 11.** Normalized FWM signal waveforms having minimum pulsewidth (a) and the frequency shift (b) at position D in Fig. 7 after the pulse compression by the fiber. This figure also shows those of the input pulse at position A in Fig. 7, which correspond to the case that the SOA acts as the ideal phase


Time (ps)

1 ps at = 0 and the shortest FWM signal pulsewidth was obtained for

**Figure 12.** Spectra and phase in the frequency domain of the FWM signal pulse at the output end of the dispersion compensation fiber shown as the position D in Fig. 7, together with those for the ideal case.

The spectra and phase in the frequency domain of the FWM signal pulse at the output end of the dispersion compensation fiber, shown as the position D in Fig. 7, are shown in Fig. 12 with the spectra and phase for the ideal case. For the 10 ps pump pulsewidth, the spectrum is almost identical to the ideal case except for the center frequency. The red shift of the center frequency originates from the gain spectrum of the SOAs. For the shorter pump pulse of 1 ps, the spectral width increased because the pulsewidth of the FWM signal becomes short, as shown in Fig. 11(a). From the figure, the output signal phase becomes nearly constant when the input pump pulsewidth is 10 ps. As a result of this simulation (modeling results), we can conclude that pump pulses longer than 10 ps are needed in order to obtain nearly ideal optical phase-conjugate characteristics for the ~2.2 ps chirped pulse.
