## 3 Results
When the length of pump stripe is shorter than 4 mm, the pump energy density is lower than the pump threshold. Therefore, the random laser action cannot occur. Figure 4: (**a**) The texture of the sample measured by an orthogonal polarizing microscope. (**b**) Schematic illustration of the random laser of the NPDDNLC sample. Figure 5: (**a**) The emission spectrum of the NPDDNLC sample as a function of the length (L) of pump stripe, when the pump energy density is 0.52 mJ/cm\({}^{2}\). (**b**) The pump threshold of the NPDDNLC sample as a function of the length of pump stripe. Here, the influence of the width of pump area on the emission spectrum is studied. In this process, the silt is removed. The pump area is an ellipse whose length (i.e., the length of the ellipse's long-axis) is fixed (about 10 mm), and its width (i.e., the length of the ellipse's short-axis) can be tuned by changing the distance between the sample and the cylindrical lens. Figure 6a shows the emission spectrum of the NPDDNLC sample at different widths of pump area when the pump energy density is about 0.54 mJ/cm\({}^{2}\). According to common sense, the number and the intensity of the discrete sharp peaks should increase with the increase of the pump area because of the increase of the effective region of the system. However, according to our experimental data, the number and the intensity of the observed discrete sharp peaks decrease with the increase of the width of pump area. As we can see in Figure 6a, when the width is shorter than 2 mm, the discrete sharp peaks can be observed clearly. When the width of pump area increases to be larger than 3 mm, the discrete sharp peaks can be observed poorly. Further increasing the width of pump area to be longer than 6 mm, it is difficult to see the discrete sharp peaks. This can be explained by two different reasons: (i) when the width of pump area increases, the light emitted from dye molecules can be amplified not only along the direction of the long side of the pump area, but also along the other directions of the pump area. The number of local modes increases with the increase of the pump area. However, the gain efficiency of the DCM molecules is limited when the pump energy density is fixed. Therefore, the fierce competition between different modes exists. In addition, the emission spectrum along other directions is not collected because the experimental data are collected along the direction of the long side of the pump area. When the gain is occupied by the modes along other directions, the intensity of the lasing modes along the direction of the long side of the pump area decreases, even these modes cannot be excited. (ii) The lasing threshold of different local modes is very similar because the random system used in our experiment is a weak scattering system [40]. When the pump area is small, fewer modes can be excited, and the distance between two adjacent lasing modes is big enough to let us observe clearly some discrete sharp peaks on the emission spectrum. When the width of pump area increases, the lasing threshold of the sample decreases due to the fact that a bigger gain area results in more modes which can be stimulated simultaneously [29]. The resonant modes are coupled to each other and their spectrum will overlap [41]. Therefore, the lasing modes cannot be distinguished. In this case, the emission spectrum is smooth and its characteristics are more like an amplified spontaneous emission (ASE). The visibility of laser modes can be evaluated by the sharpness (\(S\)) of laser modes. The sharpness of laser modes can be expressed as:  
\[S=\frac{I_{pn}-I_{sn}}{I_{pn}} \tag{1}\]  
where \(I_{pn}\) expresses the peak intensity of the nth laser mode, \(I_{sn}\) expresses the intensity of the amplified spontaneous emission (ASE) curve of the nth laser mode as shown in the inset of Figure 6a. When the  
Figure 6: (**a**) The emission spectrum of the NPDDNLC sample at different widths (W) of pump area. The inset shows the enlarged view of the corresponding rectangle area.