# Random Laser Action in Dye-Doped Polymer Media with Inhomogeneously Distributed Particles and Gain
## 3 Results
### Incoherent Random Lasing  
We first examined the emission properties of BS RLs pumped with the circular excitation pattern. Figure 3 shows examples of single-shot emission spectra obtained from random gain media, with SPs (Figure 3b-d) and without them (Figure 3a). In addition, we measured the emission spectrum from samples with SPs and without scattering particles. An ordinary fluorescence spectrum was obtained, which was approximately the same as that measured without scatterers and SPs (K40 polymer with R6G only). This indicates that the SPs themselves do not produce laser action. We observed that incoherent random lasing occurred in all samples. As discussed later, the spectral peak becomes higher and its width becomes narrower when an appropriate volume fraction of SPs is added to the RL media. The dependence of the spectral peak intensity on the pump energy is shown in Figure 4a. The slope efficiency of the BS medium is higher than that of the ordinary homogeneous medium. However, the lasing thresholds of the BS and homogeneous media are not noticeable. The lasing threshold can be identified in Figure 4b, where the spectral width is plotted as a function of the pump energy. The lasing thresholds for the BS and homogeneous media were in the range of 1.0-1.3 mJ/mm\({}^{2}\). For all pump energies, the spectral width is narrower for the BS medium than for the homogeneous medium. Figure 3: Emission spectra obtained with the circular excitation pattern for SP volume fractions of (**a**) 0%, (**b**) 5%, (**c**) 10%, and (**d**) 15%. The volume fraction of scattering particles is 10%. The pumping energy and energy density are 4 mJ and 1.3 mJ/mm\({}^{2}\), respectively. Figure 2: Schematic of the experimental setup for measuring random laser (RL) emission. L1 indicates a plano-convex lens or a plano-convex cylindrical lens, and L2 indicates a plano-convex lens. There are two possible reasons for the increase in the intensity and for the spectral narrowing: (a) the difference in scatterer distributions and (b) the difference in scattering strength. We measured the transport mean free path (TMFP), \(l^{*}\), for all samples, which is the inversely proportional to the scattering strength, using enhanced backscattering experiments [19]. Figure 5a shows the average value of the spectral peak intensity as a function of the TMFP. A high spectral peak was obtained at the TMFP in the range of 6-8 \(\upmu\)m. When \(\phi_{\text{sc}}\) is 10% (circles), the peak intensity for the BS media is \(\sim\)30% higher than that for the homogeneous medium, even if the scattering strength is approximately the same. This result indicates that the enhancement of the spectral peak intensity results from the structural difference in the random media, and not from the scattering strength. The peak intensity of the BS media is also high for the \(\phi_{\text{sc}}\) value at 5% (squares); however, the TMFP for the BS media is shorter than that for the homogeneous media. The opposite tendency is observed when \(\phi_{\text{sc}}\) is increased to 15% (triangles). Therefore, there is an optimal concentration of scattering particles for the SPs that contribute to the increase in intensity. Figure 5b shows the relationship between the average spectral width and the TMFP. The spectral width is evidently independent of the TMFP. It was observed that all the BS media exhibit narrower spectral peaks when compared with the homogeneous media. As similar to the case of the peak intensity, the spectral width decreases by \(\sim\)15% without significantly changing the TMFP when 10% SPs  
Figure 4: (**a**) Peak intensity of the emission spectra versus pump energy. (**b**) Full width at half maximum of emission spectra versus pump energy. The black rhombuses and red circles represent the homogeneous and BS media, respectively. The volume fraction of scattering particles is 10%, and the volume fraction of SPs in the BS medium is 10%. Figure 5: (**a**) Spectral peak intensity and (**b**) spectral width versus the transport mean free path (TMFP).