Chromatic dispersion spatially separates white light into colours, producing rainbows and similar effects. Detrimental to imaging but essential to spectroscopy, chromatic dispersion is the result of material properties in refractive optics and is considered an inherent characteristic of diffractive devices such as gratings and flat lenses. Here, we present a fundamental relation connecting an optical system’s dispersion to the trajectories light takes through it and show that arbitrary control over dispersion may be achieved by prescribing specific trajectories, even in diffractive systems. Using cascaded metasurfaces (2D arrays of sub-micron scatterers) to direct light along predetermined trajectories, we present an achromatic twisted metalens and experimentally demonstrate beam deflectors with arbitrary dispersion. This new insight and design approach usher in a new class of optical systems with wide-ranging applications.

色散在空间上将白光分成各种颜色，从而产生彩虹和类似的效果。色散对成像不利，但对光谱学必不可少，它是折射光学材料特性的结果，被认为是光栅和平板透镜等衍射器件的固有特性。在这里，我们提出了将光学系统的色散与光通过的轨迹联系起来的基本关系，并表明，即使在衍射系统中，也可以通过规定特定的轨迹来实现对色散的任意控制。使用级联的超表面（亚微米散射体的2D阵列）将光沿着预定轨迹引导，我们呈现了一种消色差扭曲金属元，并通过实验证明了具有任意色散的光束偏转器。