  The band structures of the photonic crystal slabs play a significant role in
manipulating the flow of light and pre-dicting exotic physics in photonics. In
this letter, we show that the key features of photonic band structures can be
achieved experimentally by the polarization- and momentum-resolved
photoluminescence spectroscopy utilizing the light emission properties of SiNx.
The two-dimensional spectra clearly reveal the energy-momentum dispersion of
band structures which is in perfect agreement with the simulation results. The
isofrequency contours can be measured easily by adding a bandpass filter with a
desired photon energy. Furthermore, it is convenient to observe clearly and
directly the optical singularity -- the optical bound states in the continuum
featured by dark point in three-dimensional photoluminescence spectra. The
polarization-resolved isofrequency contours clearly show that this dark point
is the center of an azimuthally polarized vortex. Finally, the helical
topological edge states can be easily observed in photonic topological
insulators with deformed hexagonal lattices. Our work provides a simple and
effective approach for exploring topological photonics and other intriguing
phenomena hidden in the photonic crystal slabs.
