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Fractional-order vector vortex beams are recently demonstrated to be new carriers of fractional-strength optical vortices. However, why can those new vortex beams formed by the combination of both unstable states propagate stably in free space? Here, we solve this scientific problem by revealing the physical essence of propagable fractional-strength optical vortices in free space.Three new understandings regarding those peculiar vortex beams are therefore proposed, namely Abbe diffraction limit, phase evolution of vortex beam, and phase binary time vector property.For the first one, owing to Abbe diffraction limit, the inherent polarization modes are intertwined together, thereby maintaining the entire peculiar vortex beams in free space. For the second one, we demonstrate the phase evolution of vortex beam, which is the physical reason of polarization rotation of fractional-order VVBs. For the third one, the phase is not merely a scalar attribute of light beam, but manifests a binary time vector property. This work provides entirely different physical viewpoints on the phase of vortex beam and Abbe diffraction limit, which may deepen our knowledge on the behavior of light beam in classical optics.

Is it possible to modulate the inherent properties of a single light beam, namely amplitude, phase and polarization, simultaneously, by merely its phase? Here, we solve this scientific problem by unifying all these three properties of a single light beam using phase vectorization and phase version of Malus's law. Full-property spatial light modulator is therefore developed based on the unification of these fundament links, which enables pixel-level polarization, amplitude and phase manipulation of light beams in a real-time dynamic way. This work not only implies that the amplitude, phase and polarization of a single light beam are interconnected, but also offers a solid answer on how to modulate these three natures of a single light beam simultaneously, which will deepen our understanding about the behavior of light beam, and facilitating extensive developments in optics and relate fields.

Light beam with optical vortices can propagate in free space only with integer orbital angular momentum. Here, we invert this scientific consensus theoretically and experimentally by proposing light beams carrying natural non-integer orbital angular momentum. These peculiar light beams are actually special solutions of wave function, which possess optical vortices with the topological charge l+0.5, where l is an integer. Owing to the interaction of phase and polarization singularity, these vortex beams with fractional topological charge can maintain their amplitude and vortex phase even when they propagate to an infinite distance. This work demonstrates another state of optical vortices in free space, which will fundamentally inject new vigor into optics, and other relate scientific fields.

Vector polarization induced by a change in scalar phase has been far beyond our understanding of the relationship between polarization and phase in classical optics due to the entanglement of inherent polarized modes of light beams. To overcome this limitation, we establish this miraculous relationship by the principle of phase vectorization, which can transform three particular phases into linear, circular and elliptical polarizations. A polarized-spatial light modulator based on the principle of phase vectorization can be achieved using a phase-only spatial light modulator, which not only enables pixel-level polarization manipulation of light beams in a real-time dynamic way but also simultaneously retains complete phase control. This work demonstrates the phase-to-polarization link and reports the creation of polarized-spatial light modulator, which will transform our view of the natural properties of light beam and pave the way for the era of vector optics.

The dielectric microsphere can be directly embedded into the traditional microscope, which can significantly improve the resolution of the microscope and provide a simple and feasible way to break through the diffraction limit of optical imaging system. However, due to the lack of effective theory and formula system, the resolution limit, magnification and optimal imaging position of the microsphere microsystem cannot be determined, which hinders its application. In this paper, the microscopic theory of dielectric microspheres is studied systematically, and the ray optics is extended to the imaging of dielectric microspheres, so as to establish a formula system containing important optical parameters such as resolution, magnification and imaging position, which provides a solid theoretical basis for further optimization of the nano-microscopic properties of dielectric microspheres. Compared with simulation and experiment, the correctness of the formula system is ensured. The formula shows that the mismatch between refractive index and ambient refractive index limits the resolution of microspheres, and it is an effective method to further improve the resolution to find the optimal refractive index ratio. In addition, the larger refractive index of the medium microsphere and the refractive index of the environment are conducive to the enhancement of the imaging magnification, but the working distance of the microscope objective must be taken into account in the experiment to obtain the ideal imaging position. The theoretical system proposed in this paper effectively explains the optical principle of dielectric microsphere imaging, gives optimized optical parameters, and has been applied in experiments, which is of great value for the practical application of dielectric microsphere nanometer microscopy., Comment: 10 pages, 5 figures 2 tables