## 3 Results and Discussion
In the range from 38 mW to 72 mW, the laser with 10% front facet reflectivity shows a multi-mode operation (corresponding to injection currents from 110 mA to 170 mA, see Figure 2c). For this reason, the spectral linewidth of the device was not evaluated in this range. The laser with 30% reflectivity was investigated from 24 mW to 86 mW output power and exhibited linewidths of 27 kHz and 8 kHz, respectively. The linewidth reduction factor in this case is 3.4 (power ratio 3.6). By fitting the results with a 1/P dependence, we get the linewidth power products 2274 Hz\(\cdot\)W (for R \(<\) 0.01%), 1574 Hz\(\cdot\)W (for R = 5%), 1014 Hz\(\cdot\)W (for R = 10%), and 659 Hz\(\cdot\)W (for R = 30%). As can be seen in the inset of Figure 4, the lasers with R = 5% and R = 10% reach the same linewidth at their highest single-mode operation injection currents. The device with R = 5% reaches the linewidth of 8 kHz at an output power of 290 mW and the other one at an output power of 215 mW. With increasing power, the linewidth of the laser with R \(<\) 0.01% continues to reduce to 17 kHz at 240 mW output power. To explain the influence of the front facet reflection on the electro-optical properties as well as the spectral linewidth, we calculated the external differential efficiency and the linewidth power product of the lasers according to References [24; 25]. The calculation was performed in three steps. First the carrier-density dependent part of the complex dielectric function and the rate of spontaneous emission of the GaAsP QW was calculated as a function of carrier density and wavelength using an 8 \(\times\) 8 kp band model as described in References [26] and the results are stored in a look-up table. The imaginary part of the dielectric function gives the optical gain. The ratio of the derivatives of the real and imaginary parts of the dielectric function with respect to the carrier density is the linewidth enhancement factor [24]. In the second step the complex waveguide equation and a lateral diffusion equation were solved for the given applied voltage, power and wavelength, using the look-up table  
Figure 3: Frequency noise PSD (power spectral density) of the laser beat note signal in a self-delayed heterodyne setup at 85 mW output power for lasers with various front facet reflection: R \(<\) 0.01% (red solid); R = 5% (blue dashed); R = 10% (pink dot-dashed); and R = 30% (green short dashed). created in the first step. The results such as injection current, effective index, modal gain and functions needed to calculate the spectral linewidth are stored in a table. Lastly, the electro-optical characteristics as well as the spectral linewidth are calculated by solving the coupled wave equations using the look-up table generated in the second step [25]. In this way, both lateral and longitudinal spatial hole burning have been taken into account. The parameters used in the simulation have been largely taken from Reference [27]. A summary of the most important parameters is given in Table 1. The reflection coefficient of the rear facet was set to 95%. The front facet reflections were 0.0001%, 5%, 10%, and 30%. The simulation scheme sketched above includes the dependence of effective linewidth enhancement, population inversion and the Petermann \(K\) factors on the reflections at the front facet and the phases of the grating at the rear and front facets. To compare with the experimental results the grating (or facet) phases have been varied between 0 and 2\(\pi\). The minimum and maximum values of the external differential efficiency and the spectral linewidth due to a spontaneous emission noise at an output power of 50 mW have been determined [28].