Due to the frequency dependence of light propagation in the medium, the dispersion phenomenon is widespread in nature. Dispersion can be tuned by a two-dimensional planar optical device consisting of subwavelength structures called metalens [[1], [2], [3], [4], [5], [6]]. By manipulating the dispersion states of metalenses [[7], [8], [9], [10]], researchers have demonstrated their ability to achieve diverse functions [[11], [12], [13], [14], [15]]. Achromatic metalenses can be used to improve imaging quality by reducing dispersion [[16], [17], [18], [19], [20], [21], [22], [23], [24]], and metalenses with large dispersion have important applications in fields such as spectroscopy [25]. Nevertheless, the aforementioned metalenses demonstrate a fixed dispersion state once they are fabricated. Recently, integrating different functions into a single device has become an ingenious and emerging approach to increase the optical integration and multifunctionality of metalenses [26,27]. Tunable multifunctional metalenses based on this idea have been extensively studied [[28], [29], [30], [31], [32], [33]]. Integrating metalenses with diverse dispersion states facilitate more complex and diverse applications, surpassing the functions attainable with a single-dispersion-state metalens. For example, a metalens with dispersive and hyper-dispersive states can adjust the depth of fast spectral tomography [34]. Metalenses capable of achieving achromatic and dispersive states can be used in the biomedical field for both cellular imaging [35] and coherence tomography [34]. In addition, these metalenses can also expand functionality of the systems like simultaneous implementation of normal and 3D imaging [36]. These solutions will undoubtedly reduce the complexity of optical systems and improve the integration of devices. It would greatly promote the practical applications of spectroscopy and imaging systems.

Achieving an active tunable multi-dispersive state is introducing an additional degree of freedom to adjust the dispersion. Two different discrete bands can be used as a degree of freedom to obtain different dispersion states. But discrete wavelengths may limit the bandwidth of the metalens [37]. The liquid crystal, which changes its direction of rotation under electrical drive, can also be a degree of freedom to achieve both achromatic and diffractive (positive dispersion) states [38]. Actually, changing the refractive index of materials can also increase the freedom degrees. When the refractive index of material changes, the phase and group delay also change. Therefore, phase change materials with variable refractive index can be used to actively regulate dispersion to achieve a wider range of dispersion states. Our group has proposed active band-tunable achromatic metalenses using phase change material [39]. This indicates that it is feasible to associate phase change material with metalenses of diverse dispersion states.

At this time, a new family of reversible phase change material, namely Sb2S3, that can introduce an additional degree of freedom for dispersion modulation through a shift in the real part of the refractive index. We take advantages of the properties of Sb2S3 to design actively dispersion-dynamic metalenses in the band from 1300 nm to 1700 nm. To confirm that multiple dispersion states can be arbitrarily combined two-by-two in a single metalens, three different functional metalenses are designed. Metalens 1 (M1) exhibits diffractive and achromatic focusing, Metalens 2 (M2) displays a refractive (negative dispersion) and achromatic focusing and Metalens (M3) has diffractive and hyper-dispersive focusing, when Sb2S3 is in the crystalline and amorphous states, respectively. This approach offers new possibilities for controlling multiple dispersion states and dynamic metalenses.