**4. Chaos by injection locking**

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photon number as:

*N*

frequency. The response is written as:

optical injection will be presented.

*Rinj* 

optical injection formula can be derived as [16]:

between the slave laser and the master laser.

Substituting Eqs.(8), (9), and (10) into the rate equations (5) & (6) and omitting the small

*<sup>d</sup> N G GP P G P N*

Eliminating from the above equations we get an equation for the modulation in the

<sup>2</sup> 2 *R N d d <sup>P</sup> P P G PJ*

 

*d N <sup>P</sup> GP P G P N*

This resonance frequency is for free-running semiconductor laser, i.e., without optical injection. The modulation response can be found from Eqs. (7) and (8) by assuming an exponential solution for both and as , with is the modulation

*P*

*N* exp( ) *i t*

2 <sup>1</sup> <sup>2</sup> <sup>2</sup> 2 2 22

*R*

4

 

> 

2 *N R*

 *P* 

*G*

Now, the resonance frequency and frequency response of semiconductor laser with strong

*R*

To estimate the modulation response of optically injected semiconductor laser, the rate equations have to be solved again with strong optical injection, a resonance frequency with

> 2 2 2 <sup>1</sup> <sup>2</sup> sin *inj*

 

*inj*

*P P*

where is the resonance frequency with optical injection, and is the is phase difference

*in*

With given theoretical analysis above, the transfer function for optically injected laser was obtained and plotted in the above figures. The frequency response of the slave laser with

*p N*

*P N <sup>p</sup> <sup>p</sup>*

2

 

1

 

*p*

 

  

  (11)

(13)

(14)

(12)

(15)

(16)

quantities of the second and higher order terms, a differential equations result as:

<sup>0</sup>

where the is the damping factor and is the resonance frequency given as:

*R*

*dt* 

*dt <sup>P</sup>*

2

*dt dt* 

> *P*

( )

*R R*

 *H*

The chaotic behavior of the slave laser under strong optical injection will be given in terms of the important parameters of locking regimes which are the frequency detuning and injection ratio. The optical chaos have been observed, as we will see later, and developed through period-doubling, i.e., route-to-chaos. The analysis will show that the chaotic behavior is dependent on the injection strength and frequency detuning. The bandwidth of semiconductor laser will be verified and enhanced by optical injection. The bandwidth of such chaotic laser transmitter is enhanced roughly three times by optical injection compared with the bandwidth when there is no optical injection [14]. Chaotic dynamics and period doubling were observed experimentally in the VCSELs lately [15], in addition to edge emitting lasers.

The differential rate equations (1)-(3) were subjected to numerical solutions using fourth-order Runge-Kutta algorithm with strong optical injection and the chaotic output of the slave laser is shown in figure 6. This output was originated from period-doubling route-to-chaos. This chaotic behavior is optical injection locking parameters dependent, i.e., injection strength, and frequency detuning. This is illustrated in the following diagram.

Figure 7 shows the chaotic dynamics of the slave laser under strong optical injection, with the same parameters as in the previous figure. The chaos is a phenomenon of the generic properties of the extra degree of freedom introduced by the optical injection in rate equations. The scenario of period doubling route-to-chaos was initiated by excitability of the chaotic attractor. This attractor triggers the system in deterministic sequence reaching the chaos. This observation of bifurcation in the laser power output (the response system) related to the stimulus (the drive) when coupled, synchronization is established. The ability of such synchronizing system offers the opportunity as a chaotic transmitter in optical communications. Any further increase in the injection strength will eventually be an extreme output of chaotic dynamics.

**Figure 6.** Time series of chaotic photon number of the slave laser at detuning (--7.962 GHz), linewidth enhancement factor ( =3), and injection strength (-22 dB). 

**Figure 7.** Phase portraits of chaotic output of the slave laser as a function of injection strength.

Injection strength plays a great role in the chaotic behavior of the slave laser through perioddoubling route-to-chaos. This chaotic dynamics, when explained by nonlinear bifurcation theory, is established as the relaxation oscillations (RO) become undamped via Hopf bifurcations. This bifurcations originating from the undamped RO for Fabry-Perot lasers, while the enhanced RO damping of quantum dot lasers results in the removal of these chaotic regions [18]. Period doubling in the above figure has predicted the dependence on the injection parameters. Recently, the origin of such periodicity was explained as the beating between two wavelengths, namely, the injected wavelength and the cavity resonance wavelength [19].

The above theoretical results and predictions of chaotic dynamics agree well with the experiments on a qualitative level [20]. Many theoretical and experimental studies revealed the necessity in viewing the dynamics from a broad perspective [21-25].

#### **5. Frequency chirping**

Frequency chirping in semiconductor lasers can be suppressed by strong optical injection and hopefully this laser could have better modulation characteristics than free-running laser. As the carrier density increases, resulting from the current injection into the active region will change the refractive index of the region and generate the frequency chirping. This phenomenon can have a considerable limitations on the modulation of semiconductor laser at high bit rate. An important figure-of-merit for chirp is the chirp-to -power ratio (CPR), which is defined as the ratio of lasing frequency deviation to power deviation [26], and is defined as [27]:

$$CPR = \frac{1}{2\pi H(\alpha)} \left| \frac{d\theta}{dt} \right| \equiv \frac{\alpha}{2\pi} \alpha \sqrt{\frac{\alpha^2 + \left(U - V/\alpha\right)^2}{\alpha^2 + \left(U + V\alpha\right)^2}}\tag{17}$$

Where

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extreme output of chaotic dynamics.

related to the stimulus (the drive) when coupled, synchronization is established. The ability of such synchronizing system offers the opportunity as a chaotic transmitter in optical communications. Any further increase in the injection strength will eventually be an

**Figure 6.** Time series of chaotic photon number of the slave laser at detuning (--7.962 GHz), linewidth

**Figure 7.** Phase portraits of chaotic output of the slave laser as a function of injection strength.

enhancement factor ( =3), and injection strength (-22 dB).

$$U = \frac{1}{\tau\_{in}} \left| \frac{P\_{in}}{P} \right| \cos \phi\_L \tag{18}$$

$$V = \frac{1}{\tau\_{i\epsilon}} \left| \frac{P\_{i\epsilon j}}{P} \right| \sin \phi\_L \tag{19}$$

with is the phase of the intracavity laser field relative to the injection field. This equation provides with the fact that CPR is dependent entirely on the modulation response and the phase. The stated above equations of the injection-locked semiconductor explains the chirp suppression or reduction and its dependence on injection parameters, with this arguments it can optimize the performance of the laser. *L*

Chirp-to-power ratio of injection locked semiconductor at locking range of stable operation and as a function of modulated frequency is shown in figure 8, three different injection strength were taken and for two values of linewidth enhancement factor (α). This factor has influenced the frequency chirping characteristics of the laser under direct current

modulation, since this factor plays great role in the refractive change with injected carriers. This influence can be realized from the relation stated in Eq. (17).

The frequency chirping when considered in the output power of the slave laser, the detuning was assumed to be constant, but it is dependent on the optical input power, hence the CPR has to take into account this dependent. Also, the laser cavity frequency , is carrier dependent. So the effects of variations in frequency detuning, and laser cavity frequency have to be included in the CPR. ( ) *<sup>o</sup> N*

Frequency chirping had been decreased with increasing injection strength and with lowered value of linewidth enhancement factor. The simulation reveals the dramatic influence of injection locking on the frequency chirping characteristics. Dynamical properties have been found, experimentally lately [28], to depend on the injection strength through the evolution of the optical and electrical spectral distribution. Substantial reduction in frequency chirping was observed in the direct modulation injection-locked laser, and this reduction was much more pronounced at low modulation frequency in experiment [29].

**Figure 8.** Frequency chirping characteristics of injection locked semiconductor showing its dependence on injection strength and linewidth enhancement factor.
