As a layer of artificially designed two-dimensional planar structure, the metasurface provides a new platform for the miniaturization and integration of optical devices. In recent years, with the continuous development of this field, a variety of optical mechanisms and optical functional devices based on metasurfaces have been proposed. Despite their seemingly diverse functionalities, they can be all attributed to the control of different degrees of freedom in the Jones matrix. The Jones matrix is commonly employed to describe the ability of optical devices to control polarization, amplitude, and phase of light, with a maximum number of eight degrees of freedom. Especially, more controlled degrees of freedom lead to diverse functionalities that can be achieved. For example, a single degree of freedom in the Jones matrix can be adopted for anomalous transmission. By increasing to two degrees of freedom, such as independent control of amplitude and phase of a specific component of the Jones matrix, the integration of color printing and holography can be realized. From the perspectives of the degrees of freedom in the Jones matrix, we classify and summarize the designs and applications of metasurface research in recent years to help researchers better understand the physical mechanisms of different functionalities of metasurfaces.

Our study focuses on the designs and applications of metasurfaces from the perspectives of the degrees of freedom in the Jones matrix. We firstly argue that all the functionalities of metasurfaces can be categorized into different degrees of freedom in Jones matrix and the more controlled degrees of freedom lead to diverse functionalities that can be realized (Fig. 1). Each component of the Jones matrix has two terms of amplitude and phase. Therefore, different mechanisms to control the phases including the geometry phase, resonance phase, and propagation phase are introduced to realize one degree of freedom (Fig. 2), which can be utilized for functionalities of metalens, hologram, and anomalous transmission. By changing the size of nanorods or nanodisks, the amplitude can also be controlled. Next, we show how to employ a simple structure of nanorod to construct multiple degrees of freedom in Jones matrix, including two (Fig. 3), three (Fig. 4), four (Fig. 5), six (Fig. 6), and eight (Fig. 7). Meanwhile, the possible applications are provided.

We categorize and summarize the design methods of metasurfaces with different optical functionalities based on their degrees of freedom in the Jones matrix to provide different perspectives for the metasurface field. Although the highest number of degrees of freedom of eight in the Jones matrix has been realized, the following points can be explored. First, new optical multifunctional devices should be designed by integrating various functionalities based on the multi-degrees of freedom metasurfaces. Second, the optical performance of metasurface devices with multiple degrees of freedom should be improved. Third, the Jones matrix should be extended to the wavelength dimension to enable multi-wavelength and multi-degrees of freedom control of light fields. With continuous research and deepening exploration, the field of metasurfaces will advance with a wider range of practical applications.