Topological P-CLC superlattices and emission spectrum
Liquid crystals are an important area in the field of soft matter photonics, and it is especially attractive to break the inversion symmetry by utilizing chiral materials such as CLCs [52-54]. Our assembly belongs to the category of one-dimensional (1D) coupled arrays forming the well-celebrated SSH lattice [39,40], which incorporates a CLC possessing a spontaneously-formed periodic helical structure. Such CLC structures exhibit a photonic bandgap for circularly polarized light, with helicity matching the CLC. The spectral properties of this photonic bandgap can easily be modulated by external stimuli, such as an electric or magnetic field, pressure, light, and temperature. Therefore, tunable lasers can be readily realized based on CLCs [55, 56]. Here we employ such unique properties of CLCs to realize tunable topological lasers at low cost. We analyze the coupling of defect modes in the P-CLC superlattices and demonstrate that the mini-bands can be designed in the bandgap of the CLC, which has been proposed to achieve visible defect mode lasing with a low threshold . In experiments, we apply an out-of-plane liquid crystal orientation technology based on the femtosecond laser direct writing (FLDW) method recently developed by our group . The corresponding P-CLC superlattices benefit from a combination of optical activity and tunability characteristics of the CLCs, stability and processability characteristics of the polymer materials, and unique features of the topological structure.
Figure1(a) shows a schematic drawing of the corresponding topological structure and the intensity profile of the associated TIS. Specifically, the aforementioned SSD topological structure consists of parallelly oriented polymer ribbons doped with a fluorescent dye (PM597). The space between the ribbons is filled with a right-handed CLC (RHCLC) (see Materials and Methods). Surface relief gratings present on the sidewalls of the ribbons ensure that the CLC is well aligned (out-of-plane orientation technology) . Its helical axis is perpendicular to the ribbons. The helical periodicity p (pitch) of the CLC, which is determined by the concentration of the doped chiral agent and the temperature, is assumed to be p = 348 nm here. As shown in Fig 1(a, b1), the distances between the weakly and strongly coupled ribbons are chosen to be d1 = 34×(p/2) and d2 = 22×(p/2), respectively. The width of each polymer ribbon is set to be di = 2 μm, which is thick enough to give rise to two defect modes in the bandgap of the CLC . The two P-CLC SSH superlattices in either side of the interface have the same mini-band structure, but they possess different topological properties that are characterized by their winding numbers [60-63]. A topological phase transition occurs through the band inversion, resulting in two non-trivial TISs highly localized near the interface. These states emerge in the middle of two mini-bandgaps with wavelengths of 567 nm and 581 nm, as detailed in the Supplementary Materials (SM). Besides, the SSD structure also allows for trivial defect states on both sides of the mini-bands due to the high refractive index region formed by three closely spaced ribbons at the interface. These trivial defect states are located on the defect polymer ribbon right at the interface .
The transmission polarization optical microscopy images of the fabricated SSD topological superlattice are shown in Fig. 1(b2, b3). Under the crossed polarizers, optically isotropic polymer ribbons always look dark. The CLC regions between the ribbons also look dark when the crossed polarizers are oriented parallel/perpendicular to the direction of the polymer ribbons, but they become bright when the two crossed polarizers are rotated by 45°. This indicates that the surface relief structures on the sidewalls of the ribbons enable good alignment of the CLC  and that the helical axis of the CLC is perpendicular to the direction of the ribbons. Our experimental setup for excitation and characterization of the topological lasing can be found in the SM. For inducing the lasing activity, the P-CLC superlattice is excited by a Q-switched frequency-doubled Nd-YAG laser operating at 532 nm with a repetition rate of 1.0 Hz and a pulse duration of 4.0 ns. The typical emission spectrum measured in our experiments is shown in Fig. 1(c), where one can see that there exist two lasing peaks at the wavelengths of 567 nm and 581 nm, corresponding to the two in-minigap TISs.
Circular polarization and thermal tuning of topological emission
Circularly polarized light with the same handedness as the helical structure propagating along the helical axis is selectively reflected by a CLC, while the opposite circularly polarized light is not affected by the structure. Hence selective photonic bands and bandgaps are formed only for circularly polarized light with the same handedness, so do the mini-bands and mini-bandgaps in P-CLC superlattices. The chiral characteristics of the constituent molecules and the corresponding helical arrangement of the RHCLC phase in this topological structure has the inherent property of inversion symmetry breaking, which provides a very favorable means for the realization of circularly polarized lasing. As shown in Fig. 2(a), the yellow-colored laser emission was collected at an angle of 90° to the incident direction of the pump light. To verify the circular polarization state of the topological lasing, we used a quarter-waveplate with its slow axis parallel to the vertical axis in combination with a linear polarizer to realize the polarization transformation. The obtained result is shown in the inset of Fig. 2(b). It reveals that the TIS lasing is transformed into linearly polarized light oriented at an angle of -45° to the horizontal axis, proving that right-handed circular polarization is generated in the system.
The CLC used in our experiments exhibits a cholesteric to isotropic transition (clearing point) at about 30°C. The elastic constants of the CLCs increase with decreasing temperature. Consequently, the torsional torque of the chiral dopant increases, resulting in the decrease of the CLC pitch. Therefore, decreasing temperature causes a blueshift of the optical bandgap. In addition, the width of the bandgap increases (see SM for more details). Figure 2(b) shows the thermal tuning of the lasing wavelength. When the temperature is decreased from 24°C to 8°C, topological lasing with wavelengths decreasing from 581 nm, 567 nm, 556 nm to 544 nm can be excited successively due to the blue shift of the bandgap. At lower temperatures, one can find some situations in which three TIS lasing peaks exist simultaneously. It is conceivable that, if required, single-mode topological lasing can be achieved by decreasing the bandgap width of the CLC and/or by increasing the mode spacing by using narrower polymer ribbons.
Robustness against perturbations
We found that in our current fabrication method there exists about a 2% variation in the ribbon width and the spacing between the ribbons. To explore the effect of random fabrication errors on topological lasing, we excited the sample by steering the pump beam to five different interface positions. The resultant lasing spectra are shown in Fig. 3(a). They reveal that the TIS lasing always occurs and is preserved, although the lasing wavelength has a fluctuation within ±2.5 nm. This indicates that the TIS lasing is robust against the perturbations due to fabrication imperfections.
The TIS based on the SSH model is known to be robust against perturbations that respect chiral symmetry of the Hamiltonian [63-67]. The Hamiltonian of the SSH model containing an SSD by using the tight-binding method can be written as follow :
Comparison between TIS lasing and trivial defect state lasing
Low lasing threshold is indispensable for practical applications. This has motivated tremendous efforts on exploration of defect-mode lasing based on CLCs, which was found to show a lower threshold compared to band-edge lasing of CLCs [68-71]. As we know, topological boundary (edge, corner, interface) states are more localized and robust than the conventional defect modes, which entails an even lower lasing threshold [17-33]. In addition, robust topological modes can maintain a high slope efficiency and high wavelength stability in the presence of defects and disorder [17, 38]. In our work, the highly localized TISs provide a favorable condition for realizing low threshold lasing. As can be seen from the illustration in Fig. 4(a), topological lasing appears with increased pumping energy. The lasing threshold observed in our experiments is about 0.4 μJ, corresponding to a peak intensity of 722 W/mm2, which is almost four orders of magnitude lower than what has been achieved in previously reported visible topological laser . According to Fig. 4(b), a clear decrease in the emission linewidth, i.e., spectral narrowing is observed with increasing pump energy, which further verifies the lasing behavior of the topological P-CLC superlattice.
Figure 4(c) shows the calculated photon density of states for the SSD superlattice, from which one can see that there exist peaks from the two TISs (A and B) and four trivial defect states (C, D, E and F). Both TISs and the trivial defect states localize around the interface of the SSD structure, although the modal distributions of the TISs are different from those of the trivial defect states [45, 46]. The densities of states of the TISs are higher than those of the trivial defect states, which empowers the TISs lower lasing thresholds. Thus, the independent excitation of TIS lasing under low pump energy (0.4 μJ - 1.18 μJ) is experimentally realized due to their lower thresholds with respect to trivial defect states. This is shown in the inset of Fig. 4(a), though TISs and trivial defect states are both present near the interface. As the pumping energy is further increased, the trivial defect states are also excited. As illustrated in Fig. 4(d), there exist two TIS lasing peaks with wavelengths at 567 nm and 581 nm, and four trivial-defect-state lasing peaks with wavelengths at 564 nm, 570 nm, 579 nm and 584 nm. These experimental results agree well with our theoretical analyses.