The two-dimensional photonic crystal slab (PhCS) is a structure characterized by the spatial periodicity of the dielectric constant within the plane. In contrast to traditional metamaterial surfaces, the two-dimensional PhCS enables light field manipulation in momentum space based on the Fourier principle, thus achieving complex and diverse functionalities. Since modes above the light cone can radiate to the far field and possess definite polarization states, polarization is matched with wave vectors, defining polarization fields in momentum space. Various polarization singularities exist within the polarization field, such as V points and C points. Previous studies generally focus on information such as frequency and momentum, while the polarization field can reflect the topological information of the bands and provide a new dimension for light field manipulation. For example, by controlling the evolution of polarization singularities, researchers have obtained bound states in the continuum (BICs) with robust characteristic and unidirectional guided resonances (UGRs). By utilizing these characteristics, researchers have designed high-performance lasers and realized complex light field manipulation and such functionalities as optical information processing. Compared to traditional structures, the two-dimensional PhCS exhibits non-local characteristics and has significant advantages in miniaturization and integration. Thus, it holds promising prospects for device applications. Studying the evolution of the polarization field helps guide the structural design of photonic crystal slabs, which expands the applications in communication, sensing, and other fields, and provides a deeper understanding of how topological photonics is manifested in optical systems.

We start by introducing the definition of the polarization field in the momentum space of the two-dimensional PhCS and introduce the concept of polarization singularities (Fig. 1). Subsequently, an analysis is conducted from the perspective of symmetry, with the relationship between the topological charge of polarization singularities and the in-plane point group symmetry examined (Fig. 2 and Table 2). Additionally, we outline the description of the polarization field using the temporal coupled mode theory (TCMT). Furthermore, the conservation law followed by the topological charges during their evolution is discussed (Fig. 3) to detail the research on the evolution of polarization singularities based on whether the band is non-degenerate or degenerate. It is observed that non-degenerate V points correspond to BICs and are split into more fundamental C points during symmetry change (Fig. 4). The evolution of these polarization singularities is controlled by structural parameters and symmetry (Fig. 5). Degenerate V points typically correspond to band degeneracy points and are also influenced by structural parameters and symmetry (Fig. 6). Based on the evolution patterns of polarization singularities, researchers have designed robust merging BICs and utilized the topological charge to generate vortex beams and beam shifts (Fig. 7), providing significant guidance for laser design. Furthermore, by altering the out-of-plane symmetry, UGR can be achieved (Fig. 8). Additionally, appropriately designed PhCS can achieve full coverage on the Poincaré sphere and perform complex image processing tasks such as edge detection (Fig. 9).

Generally, the investigation of the polarization field characteristics of PhCS guides the design of appropriate structures and can help achieve complex and rich functionalities. Despite the presence of numerous unresolved physical issues currently, the application potential of the polarization field remains largely untapped. However, these unknowns are expected to stimulate enthusiasm for exploration and boost progress in related fields.