## 2 Step Modulation Principles  
The relaxation oscillation process can be set up by the rate equations, in which the steady and dynamic characteristics of a semiconductor laser can be described efficiently and accurately with the following single mode rate equations [16; 17; 18]:  
\[\frac{dN}{dt}=\frac{J}{eV}-\frac{g_{0}(N-N_{t})S}{1+\varepsilon S}-\frac{N}{ \tau_{N}}, \tag{1}\]  
\[\frac{dS}{dt}=\frac{\Gamma g_{0}(N-N_{t})S}{1+\varepsilon S}-\frac{S}{\tau_{P} }+\Gamma\beta\frac{N}{\tau_{N}}, \tag{2}\]  
The output power is given in terms of the photon density [19]:  
\[P_{out}=\frac{\eta_{0}Vhv}{2\Gamma\tau_{P}}S. \tag{3}\]  
where \(N\) and \(S\) are the carrier density and photon density, \(J\) is the injected current, other parameters and their typical values are shown in Table 1.  
Figure 1 shows the evolution in the carrier and photon densities during a gain switch cycle, assuming that the bias current is 0 and that the initial photon density in the laser can be ignored. A rectangular wave modulation process is given in Figure 1a: At time \(t_{0}\), a current pulse is applied to a laser with the amplitude of \(I_{0}\) and pulse width of \(T_{0}\). The initial rate of increase in photon density is very low, due to its low initial value. Without being significantly consumed by stimulated emission, injected electrons rapidly build the carrier density up to threshold \(N_{tr}\) at time \(t_{1}\). After that, the population inversion can be achieved, and the laser begins to emit the light pulse. When the  
stimulated emission begins to consume the carrier significantly, the population inversion reaches the maximum value \(N_{\text{max}}\). With the generation of laser pulses, the carrier density drops to the threshold at time \(t_{2}\), at which point the output pulse power reaches its peak value. At this time, the current should be terminated quickly to restrain the secondary oscillation. Therefore, to form a single pulse output with high peak power, the rectangular current pulse must be terminated for the first time when the carrier density drops to the steady-state threshold [20]. The above analysis shows that, before the carrier density reaches the threshold, the photons are mainly generated by the spontaneous emission of the laser. Afterward, the carriers injected into the laser begin to be converted into stimulated emission photons. The peak reversal rate of carrier density given in the Reference [21] is \(R=N_{\text{max}}/N_{tr}\), and with increasing \(R\), the peak power of the output pulse increased while the full width at half maxima (FWHM) decreased. Therefore, if we can increase the modulation current suddenly at the moment when the carrier density reaches the threshold, the peak reversal rate can be increased, and the optical pulse with higher power and shorter width will be obtained. To do this a step-pulse modulated signal is designed, as shown in Figure 1b, in which the amplitude of the first modulation current is \(I_{1}\), the pulse width is \(T_{1}\), the second modulation current amplitude is \(I_{2}\), and the pulse width is \(T_{2}\). Figure 1b shows the step-pulse modulation process: When the current is injected at time \(t^{\prime}_{0^{\prime}}\), the carrier density increases gradually and achieves the threshold at time \(t^{\prime}_{1}\); then the increase rate of carrier density is obviously improved and the peak power of output pulse increases as the modulation current becomes larger; finally the peak power reaches a maximum when the carrier density drops to the threshold at time \(t^{\prime}_{2}\).