The field of topological photonics emerged as one of the most promising areas for applications in transformative technologies: possible applications are in topological lasers or quantum optics interfaces. Nevertheless, efficient and simple methods for diagnosing the topology of optical systems remain elusive for an important part of the community. Herein, a summary of numerical methods to calculate topological invariants emerging from the propagation of light in photonic crystals is provided. The fundamental properties of wave propagation in lattices with a space‐dependent periodic electric permittivity is first described. Next, an introduction to topological invariants is provided, proposing an optimal strategy to calculate them through the numerical evaluation of Maxwell's equation in a discretized reciprocal space. Finally, the tutorial is complemented with a few practical examples of photonic crystal systems showing different topological properties, such as photonic valley‐Chern insulators, photonic crystals presenting an “obstructed atomic limit”, photonic systems supporting fragile topology, and finally photonic Chern insulators, where the magnetic permeability is also periodically modulated.

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In recent years, there have been rapid advances in the parallel fields of electronic and photonic topological crystals. Topological photonic crystals in particular show promise for coherent transport of light and quantum information at macroscopic scales. In this work, we apply for the first time the recently developed theory of "topological quantum chemistry" to the study of band structures in photonic crystals. This method allows us to design and diagnose topological photonic band structures using only group theory and linear algebra. As an example, we focus on a family of crystals formed by elliptical rods in a triangular lattice. We show that the symmetry of Bloch states in the Brillouin zone can determine the position of the localized photonic wave packets describing groups of bands. By modifying the crystal structure and inverting bands, we show how the centers of these wave packets can be moved between different positions in the unit cell. Finally, we show that for shapes of dielectric rods, there exist isolated topological bands which do not admit a well-localized description, representing the first physical instance of "fragile" topology in a truly noninteracting system. Our work demonstrates how photonic crystals are the natural platform for the future experimental investigation of fragile topological bands.Introduction. In recent years, there have been tremendous parallel advances in the fields of both topological electronic materials and engineered photonic crystals. On the one hand, topologically nontrivial band insulators have been discovered [1-6] which feature protected, gapless surface, edge, and hinge [7][8][9][10][11][12][13] states, as well as anomalous bulk response functions [14][15][16]. The interplay between topology and crystal symmetry in these phases has reinvigorated the study of band theory, and resulted in new connections between topology in momentum space and the real-space orbital structure of electronic solids [17][18][19][20][21]. Following Haldane and Raghu's seminal ideas [22,23], many of these concepts have also been simultaneously explored in the propagation of photons in periodic dielectric structures (photonic crystals). For instance, photonic analogs of the quantum Hall effect [24,25], quantum spin-Hall effect [26-28], quantum valley-Hall effect [29], Floquet topological insulators [30-33], mirror-Chern, and quadrupole insulating [34-42] systems have been recently discovered.Because photons in linear dielectrics are truly noninteracting, and since they can be cheaply and easily engineered with almost any desirable lattice structure, two-dimensional * aitzolgarcia@dipc.org †

Time Reversal Symmetry (TRS) broken topological phases provide gapless surface states protected by topology, regardless of additional internal symmetries, spin or valley degrees of freedom. Despite the numerous demonstrations of 2D topological phases, few examples of 3D topological systems with TRS breaking exist. In this article, we devise a general strategy to design 3D Chern insulating (3D CI) cubic photonic crystals in a weakly TRS broken environment with orientable and arbitrarily large Chern vectors. The designs display topologically protected chiral and unidirectional surface states with disjoint equifrequency loops. The resulting crystals present the following characteristics: First, by increasing the Chern number, multiple surface states channels can be supported. Second, the Chern vector can be oriented along any direction simply changing the magnetization axis, opening up larger 3D CI/3D CI interfacing possibilities as compared to 2D. Third, by lowering the TRS breaking requirements, the system is ideal for realistic photonic applications where the magnetic response is weak.

√ 3 2 ) and a 3 = a 2 − a 1 = (−1, 0) the lattice vectors. The Hamiltonian (1) is expressed in the basis = {ψ A , ψ B , ψ C } T , where T represents the transposition. We show in Fig. 1(b) the band structure for a periodic lattice 085411-2