## 3 Results
In addition, the slope efficiency of the NPDDNLC sample, defined as the increase amplitude of the peak intensity of emission spectrum for unit pump energy density, is about 10 times larger than that of the DDNLC sample, which means that the conversion rate of the pump energy into lasing is increased dramatically by adding the TiN nanoparticles into DDNLC. Figure 4a shows the texture of the sample measured by an orthogonal polarizing microscope. Figure 4b shows the schematic illustration of the random laser of the NPDDNLC sample. The non-oriented cells result in the non-uniform alignment of NLC molecules. As shown in Figure 4a, the NLC molecules are self-assembled into many micro-domains and the micro-domain possesses a residual birefringence [37; 38]. The orientation of the micro-domains near the inner surfaces of the cell is parallel to the surfaces of the cell due to the weak anchoring force from the inner surfaces of the cell. The orientation of the micro-domains away from the surfaces of the cell is chaotic. Because of the long-range fluctuations of the micro-domain director, the dielectric tensor of NLC molecules is different between the adjacent micro-domains, then the spatial refractive index is not uniform in the cell [39]. The light is confined in the cell and has a longer dwelling time in the cell because of the multiple scattering of micro-domains. This is also the reason why random lasers can occur in the DDNLC samples. In addition, the introduction of TiN nanoparticles into the DDNLC solution can enhance the emission spectrum by two mechanisms. On the one hand, the electromagnetic field around TiN nanoparticles is enhanced by the LSPR of the TiN nanoparticles. This effectively increases the pump absorption and the fluorescence amplification efficiency of the DCM molecules around TiN nanoparticles. On the other hand, the scattering strength of the sample is enhanced by adding TiN  
Figure 3: **(a)** Evolution of the emission spectrum from the DDNLC sample as a function of the pump energy density. **(b)** The peak intensity and FWHM of the emission spectrum from the DDNLC sample as a function of the pump energy density. nanoparticles into the sample. This makes that the dwelling time of the light in the sample is increased markedly. When the gain exceeds the loss, the random laser occurs. In order to study the influence of the length of pump stripe on the emission spectrum, an adjustable slit is added in front of the sample as shown in Figure 1, which can change the length of pump stripe illuminated on the sample. Figure 5a shows the emission spectrum of the NPDDNLC sample as a function of the length of pump stripe, when the pump energy density is 0.52 mJ/cm\({}^{2}\). The emission intensity decreases with decreasing the length of pump stripe from 10 mm to 2 mm. In addition, the number of the discrete sharp peaks decreases markedly with the decrease of the length of pump stripe. As we can see in Figure 5a, the discrete sharp peaks disappear when the length of pump stripe decreases to be less than 4 mm. This is because the gain length of the photons emitted from the dye molecules reduces with the decrease of the length of pump stripe. The effective region of the system [40], defined as the pump region plus the absorption length, decreases with the decrease of the length of pump stripe, which makes the reduction of the random laser modes in the sample. When the length of pump stripe decreases to a given value, the discrete sharp peaks will disappear due to the fact that the gain is smaller than the loss. It also means that the pump energy is lower than the threshold. Figure 5b shows the pump threshold of the NPDDNLC sample as a function of the length of pump stripe. The pump threshold increases with the decrease of the length of pump stripe. When the length of pump stripe is longer than 6 mm, the pump energy density (0.52 mJ/cm\({}^{2}\)) is larger than the pump threshold. Therefore, the discrete sharp peaks can be observed.