**3. Bandwidth enhancement by injection locking**

The bandwidth of semiconductor laser is limited by the relaxation oscillation frequency. When a small-signal transient response is applied to rate equations of semiconductor laser as a step increment in the injection current and solved analytically, we will observe that the photon and electron numbers approach asymptotically to a new steady-state after an evanescent oscillation frequency (called relaxation) and with high damping factor. The relaxation oscillation frequency depends on the laser structure and the operation conditions. Any expression of the relaxation oscillation will show that it depends on the differential gain and the volume density of the injection current, as well as the photon lifetime. We will show the power dependence of the relaxation oscillations and the dependence on differential gain, while neglecting the gain saturation (intensity dependence) for simplicity in this monograph.

The frequency response of the intensity modulation of the laser describes the transfer of the modulation from current to optical power output. Figure 3 shows the frequency response of the semiconductor laser with and without strong optical injection. The response is flat in the low-frequency region and with a peak in the vicinity of the relaxation oscillation frequency. The response shows a rapid decrease for frequencies above the relaxation frequency. This means that the maximum frequency response of the laser when directly modulated is limited by the relaxation oscillation frequency.

Strong optical injection, as shown in figure 3, will shift the relaxation oscillation (RO) frequency (cutoff frequency) of the semiconductor to higher values depending on the operating conditions of the laser. This shift due to optical injection is effective for enhancing the modulation characteristics of semiconductor lasers. The shift is more obvious when plotted as a function of injection strength, as in figure 4. The laser was biased above threshold at 1.6 Jth and frequency detuning was of -7.962 GHz. The cutoff frequency can be said to be proportional to the injection strength, and hence modulation response of the laser can be enhanced due to strong optical injection [16].

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**Figure 2.** Locking characteristics of the slave laser showing the stable (locking) and unlocking regions

The bandwidth of semiconductor laser is limited by the relaxation oscillation frequency. When a small-signal transient response is applied to rate equations of semiconductor laser as a step increment in the injection current and solved analytically, we will observe that the photon and electron numbers approach asymptotically to a new steady-state after an evanescent oscillation frequency (called relaxation) and with high damping factor. The relaxation oscillation frequency depends on the laser structure and the operation conditions. Any expression of the relaxation oscillation will show that it depends on the differential gain and the volume density of the injection current, as well as the photon lifetime. We will show the power dependence of the relaxation oscillations and the dependence on differential gain, while neglecting the gain saturation (intensity dependence) for simplicity

The frequency response of the intensity modulation of the laser describes the transfer of the modulation from current to optical power output. Figure 3 shows the frequency response of the semiconductor laser with and without strong optical injection. The response is flat in the low-frequency region and with a peak in the vicinity of the relaxation oscillation frequency. The response shows a rapid decrease for frequencies above the relaxation frequency. This means that the maximum frequency response of the laser when directly modulated is

Strong optical injection, as shown in figure 3, will shift the relaxation oscillation (RO) frequency (cutoff frequency) of the semiconductor to higher values depending on the

for two values of linewidth enhancement factor.

limited by the relaxation oscillation frequency.

in this monograph.

**3. Bandwidth enhancement by injection locking** 

**Figure 3.** Modulation response of free-running semiconductor laser and with injection-locked laser at different bias currents . Frequency detuning was -7.962 GHz.

Modulation response of the semiconductor laser as being dependent on both the operating conditions and optical injection is illustrated in figure 5. As can be seen from the graph, the shift of RO to a higher frequencies with increasing the injection strength. The strength of RO resonance plays a crucial role in harmonic distortions. It follows that the lasers of having a high RO would result in large harmonic distortions. The resonance peaks can be explained as a result of frequency domain manifestation of the time domain of the optical field of the laser. The fall-off in the modulation response is due to a combination of the intrinsic laser response and the effects due to parasitic elements in the device.

The theory of modulation characteristics of strongly injected semiconductor laser will be stated. Modulation of semiconductor laser had been studied since the invention of this laser and the approach was using the rate equation with noise free and with periodic injection current. Steady-state solutions of the rate equations can be found for constant injection rate. Small fluctuations (first-order perturbations) terms were then added to steady-state as given below. Modulation response when evaluated can be a treated as a direct measure of the rate

at which information can be transmitted (primarily baseband). Modulation bandwidth at any biasing current or optical power is an important for optical communication systems.

**Figure 4.** Cutoff frequency dependence on the injection strength of the injection locked semiconductor laser. The solid line is the best fit of the numerical results.

**Figure 5.** Modulation response of injection locked semiconductor laser as a function of injection strength and modulation frequency. The slave laser was detuned at -7.962 GHz

Single mode rate equations of free-running laser can be regarded as either as a model for purely single-mode operation or as an approximate model for the dynamics of the total photon number. The carrier number N and photon number P can be written as:

$$\begin{aligned} \frac{dN(t)}{dt} &= \text{injected\ carrier} - \text{spont.emission} - \text{estimated\ emission} \\\\ \frac{dP(t)}{dt} &= \text{(Gain} - \text{loss)} + \text{spont.term\ coupled to\ Ising mode} \end{aligned}$$

Now, the two above equations will read as:

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at which information can be transmitted (primarily baseband). Modulation bandwidth at any biasing current or optical power is an important for optical communication systems.

**Figure 4.** Cutoff frequency dependence on the injection strength of the injection locked semiconductor

**Figure 5.** Modulation response of injection locked semiconductor laser as a function of injection

strength and modulation frequency. The slave laser was detuned at -7.962 GHz

laser. The solid line is the best fit of the numerical results.

$$\frac{dN\,\,(t)}{dt} = J(t) - \frac{N}{\tau\_{sp}} - G\,(N, P)P\,\tag{5}$$

$$\frac{dP(t)}{dt} = \left(G(N, P) - \frac{1}{\tau\_p}\right) P + \beta \frac{N}{\tau\_{sp}} \tag{6}$$

where the gain *G (N, P)* can be expressed, assuming linear gain dependence on the carriers, as:

$$G(N, P) = \frac{G\_o + G\_N(N - N\_o)}{1 + \varepsilon P} \tag{7}$$

where is the linear gain, is the differential gain at *No* , and the factor *Go o N N N <sup>G</sup> <sup>G</sup> N* 

 accounts for nonlinear gain saturation. The gain saturation becomes important at high photon numbers. The factor is called the gain suppression coefficient. 1 *P* 

 is the fraction of spontaneous emission coupled to the lasing mode. The steady-state equations can be found by setting *<sup>d</sup> dt* equal to zero. 

First-order perturbation is written as:

$$
\bar{N} = \bar{N} + \delta N \quad \text{ / } \left| \delta N \right| \ll \bar{N} \tag{8}
$$

$$\bar{P} = \bar{P} + \delta P \quad , \; \left| \delta P \right| \ll \bar{P} \tag{9}$$

Assuming the gain is a function of the carriers and photons , and can be approximated by Taylor expansion around the bias as: *N P*

$$G\left(N\right), P\right) = G\_{\ o} + G\_{\ N} \delta N\,\,\, + G\_{\ p} \delta P\,\,\,\tag{10}$$

where is the differential gain, and is the saturation gain. *GN GP*

Substituting Eqs.(8), (9), and (10) into the rate equations (5) & (6) and omitting the small quantities of the second and higher order terms, a differential equations result as:

$$\frac{d}{dt}\delta N = -\left(G\_0 + G\_p \overline{P}\right)\delta P - \left(G\_N \overline{P} + \frac{1}{\tau\_p}\right)\delta N\tag{11}$$

$$\frac{d}{dt}\frac{d}{dt}\delta P = -\left(\frac{\mathcal{J}\,\bar{\hat{N}}}{\tau\_p\,\bar{P}} - G\_p\,\bar{P}\right)\delta P + \left(G\_N\,\bar{P} + \frac{\mathcal{J}}{\tau\_p}\right)\delta N \tag{12}$$

Eliminating from the above equations we get an equation for the modulation in the photon number as: *N*

$$\frac{d^2}{dt^2}\delta P + 2\Gamma \frac{d}{dt}\delta P + \phi\_\kappa^2 \delta P = G\_N \stackrel{\cdot}{P} J \tag{13}$$

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

$$
\rho \rho\_R^2 = \frac{G\_N}{\pi\_p} P \tag{14}
$$

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 frequency. The response is written as: *P N* exp( ) *i t* 

$$\left| H \left( \phi \right) \right| = \frac{\left\| \phi \right\|\_{\mathbb{R}}^2}{\left[ \left( \phi \right)^2 - \left\| \phi \right\|^2 + 4\Gamma^2 \phi^2 \right]^{\frac{1}{2}}} \tag{15}$$

Now, the resonance frequency and frequency response of semiconductor laser with strong optical injection will be presented.

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 optical injection formula can be derived as [16]:

$$
\alpha\_{R\_{\text{ny}}}^2 \approx \alpha\_{\text{R}}^2 + \left(\frac{1}{\sigma\_{\text{in}}}\right)^2 \left(\frac{P\_{\text{ny}}}{P}\right) \sin^2 \theta \tag{16}
$$

where is the resonance frequency with optical injection, and is the is phase difference between the slave laser and the master laser. *Rinj* 

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 the highest enhanced resonance frequency can be achieved by optical injection and this found to be more than five times increased as compared to experimental results [17]. The bandwidth accordingly will be increased due to this increase in the frequency response of the slave laser. This enhancement in the bandwidth is dependent on the optical injection parameters and when the slave laser is operated in the stable locking region. Modulation enhancements characteristics of strongly injected slave laser are mainly due to the shift in the resonance frequency have to be understudied thoroughly. The theory of semiconductor laser, as a matter of fact, always simplify the arguments and make assumptions in order to make the an analytical solutions available. Numerical simulations for the laser nowadays can be found in easy used packages in the market. We have analyzed the response of semiconductor laser based on the rate equations and solved numerically. The change in the photon number due to optical injection is not the only source of increasing bandwidth but also the competition between the frequency of dominant mode of the slave laser and the frequency shift induced by strong optical injection. This bandwidth enhanced of slave laser is ultimately a candidate source for optical communications, not due to its enhanced bandwidth but also to its chaotic output as secure transmitter.
