# Timing Photonic Topological Insulator for Laser Applications
## 1 Introduction  
Recently, a topological insulator laser was demonstrated [1; 2] which exhibits scattering-free edge-state transport of light in the laser cavity, immune to perturbations and disorder. The topological feature forces injection locking of many laser emitters to act as a single coherent laser. The concept led to a variety of configurations: electrical pumping of a quantum cascade laser with valley modes [3]; vertical emission [4]; and bulk and vortex lasing [5; 6]. Simultaneously, one-dimensional active resonator arrays were described by a Su-Schrieffer-Heeger model [7], and then implemented and optimized for robust single-mode high-power lasers [8; 9; 10].  
Topological invariance of a physical process means that it is described by some highly robust conserved quantity that does not change under continuous deformations in the parametric space [11]. For example, the number of holes of a connected smooth compact surface does not change as the surface deforms, without cutting or gluing. In particular, [PERSON]'s theorem connects symmetries to the existence of the corresponding conserved quantities, such as energy, momentum, angular momentum, electric charge, and others. Parametric deformations can break the symmetry, still preserving the topological invariant. Almost forty years ago, a topological invariant was introduced for a single-particle electron state in a combined periodic potential and out-of-plane magnetic field [12; 13]. The similar wave-like behavior is inherent to the photon state in the periodic potential of the photonic crystal [14]. As the Bloch wavenumber adiabatically passes the loop trajectory within the Brillouin zone of the crystal, its wave function acquires some topological phase that defines the topological invariant of the photonic band, separated by an insulating bandgap from neighboring bands. According to the bulk-edge correspondence principle [15], the number of edge states in the bandgap is equal to the difference of topological invariants in neighboring bands. Here, one assumes the _spatial edge_ of the array rather than frequency band edge. As usual, in the optical frequency range, the magnetic response is weak. Instead, a spin-orbit coupling [16], dynamical modulation [17], or gain-loss materials [2] can also giverise to nonzero topological phases separated by an insulating bandgap. This idea was easily transferred from photonic crystal to waveguides, including ring resonator platforms [18].  
Quartz prism resonators serve well instead of ring resonators, as the excitation can propagate between prisms by the _frustrated total internal reflection_ mechanism. For the latter structure, additional edge solutions arise at different incident angles [19]. Nonetheless, there is the question of realizing such a complex coupling utilizing synthetic dimensions [20], or in the microring fashion [21,22]. If the structure period is comparable to radiation wavelength, then the description engages the topological phase. Otherwise, when the resonators are much larger than the wavelength, a phase-free explanation of the edge state is appropriate in terms of the _edge trajectory_ of the beam [19].  
In this paper, we suggest a _tiling photonic topological insulator_ (TPhTI) constructed from identical prism resonators. Compared to [19], this new design suggests a continuous connection of neighboring resonators without spatial gaps. The light beam circulates within a few prisms inside the tiling bulk, penetrating through the mutual faces of the prisms or turning the propagation direction by total internal reflection at the open faces. However, a topologically-propagating state occurs at the edge of the array due to the disconnected faces of edge prisms. This propagating state persists against removing or attaching new prism resonators.  
Furthermore, we fabricated the TPhTI and demonstrated the topologically-protected propagating state experimentally. We also speculated on laser applications of the TPhTI containing the gain material.