Influence of gradient phased interfaces on the laws of light propagation

The gradient phased interface is characterized by a non-zero phase variation along the interface between two optical media,which could generate a phase shit between the emitted and incident light beams.Unlike common ones,gradient phased interfaces have a great influence on the laws of light propagation,including light reflection and refraction,and some novel phenomena are observed.For a comprehensive understanding the optical characteristics of those gradient surfaces,the universal laws of light propagation at gradient phased interfaces are derived and discussed in detail in this paper.According to Fermat's principle,we use the stationary phase method to successively acquire the two-dimensional (2D) and three-dimensional (3D) generalized laws of reflection and refraction.In the 2D generalized laws,the interfacial phase gradient lies in the plane of incidence,which is coplanar with the incident,refracted and reflected light beams. But in the 3D case,the phase gradient does not lie in the plane of incidence,and the non-planar reflection and refraction phenomena are observed.These generalized reflection and refraction laws indicate that the interface between two media could be an important factor when light traverses it,and gradient phased interfaces provide new degrees of freedom for manipulating the wavefront of light beams.Based on the generalized reflection and refraction laws,we analyze the influence of phase gradient on light propagation,then obtain critical angles of incidence for reflection and refraction (i.e.the critical angles for total internal reflection and total transmission) in 2D and 3D cases,and explain the reasons for some novel phenomena,such as reflection angle unequal to incidence angle,anomalous reflection and refraction, out-of-plane reflection and refraction,etc.These analysis results show that generalized laws of reflection and refraction have important value in optical design.In addition,we propose an optical design idea based on generalized laws of reflection and refraction,in which gradient phased interfaces are used as core components of optical elements to perform optical transform.And then a flat lens and flat axicon are taken for example to illustrate this idea,the design process of the two flat optical elements are shown in detail.Moreover,we experimentally simulate the gradient surfaces of the two elements by spatial light modulator,and experimental results agree well with theoretical values.It proves that this design idea is practicable.Our research is useful to understand comprehensively the generalized reflection and refraction laws,and extend the applications of generalized laws to flat optics,freeform optics and the accurate control of complex wavefront.