## 3 Optically Pumped Lasers
### Distributed Feedback (DFB) Lasers  
DFB lasers are basically made by an active waveguide, in which a periodic modulation of the refractive index along the light propagation direction allows constructive interference between the partially reflected components at the wavelengths satisfying the Bragg condition \(2n_{eff}\Lambda=m\lambda\) (where \(n_{eff}\) is the effective refractive index of the waveguide, \(\Lambda\) the grating period, m the interference order and \(\lambda\) the light wavelength). The feedback for lasing comes from partial reflection at any refractive index variation, and it is thus distributed along the film. Moreover, the light propagation along the films results in large amplification, thus allowing to combine single mode operation (for small enough grating period) with efficient amplification along the active film. The refractive index periodic modulation is typically obtained by realizing periodic unidimensional gratings either on the substrate/film or on the film/air interfaces (see Figure 11a). This allows the realization of DFB lasers by simple deposition of the active film on a properly patterned substrate or by imprinting the  
Figure 10: (**a**) SiO\({}_{2}\) microsphere coated with a MAPbI\({}_{3}\) thin film. Adapted with permission from [62]. American Chemical Society, 2014. (**b**) Single mode tunable lasing in CsPbX\({}_{3}\) spheres with different chemical composition. Adapted with permission from [63]. American Chemical Society, 2017. (**c**) Optical micrograph of MAPbBr\({}_{3}\) crystals grown in a SiO\({}_{2}\) microcapillary with internal diameter of 25 \(\upmu\)m. (**d**) Cross sectional SEM image of the same sample. (**c**,**d**) adapted with permission from [64]. AIP publishing, 2018. (**e**) Lasing spectra of 8 different MAPbBr\({}_{3}\) microdisks realized by top-down lithography providing evidence of the excellent spectral reproducibility. Adapted with permission from [65]. John Wiley and Sons, 2017.  
top surface with a patterned stamp, after the film deposition, by soft lithography techniques, thus avoiding any wet lithography that can damage the perovskites layer. The first demonstration of single mode DFB optically pumped perovskite lasers was obtained by [PERSON] et al. [67] by depositing a MAPbI3 thin film from solution on the polymeric replica of a silicon master pattern, obtained by nano imprint lithography (NIL). The excitation density dependence of the emission spectra, under 1 ns pumping at 532 nm, allowed to observe the appearance of a very narrow peak at 786.5 nm, with a resolution limited linewidth of 0.2 nm, above a threshold excitation density of 120 \(\upmu\)Jcm\({}^{-2}\) (see Figure 11b,c). To unambiguously demonstrate that these features were due to lasing the authors also compared the threshold and the linewidth with the ASE ones, measured on the same film in a planar region of the substrate. They observed that the ASE linewidth and threshold were both higher than the values found in the region of the substrate pattern. Moreover the polarization of the emission was measured, demonstrating that the ASE is not polarized, while the laser peak has a clear linear polarization. After this first breakthrough the same group also realized the first flexible perovskite DFB laser by ink-jet printing deposition of MAPbI3 [71]. In this case, a pattern realized on silica was replicated by PDMS, then the PDMS stamp was used to transfer the pattern to a OrmoComp UV resist layer, deposited on a PET substrate. The MAPbI3 layer was then deposited by ink-jet printing of a precursor solution in DMSO. The final device showed again single mode lasing under 1 ns pumping at 532 nm, with a threshold of 270 \(\upmu\)Jcm\({}^{-2}\), and a linewidth of 0.4 nm, both about 2 times higher than the ones of the laser realized by spin coating.