Diffractive optical elements (DOEs) play an important role in modern optical applications such as spectral and imaging systems, but it is challenging to balance the diffraction efficiency with the working bandwidth. The core issue is controlling the broadband dispersion of all phase units to achieve achromatic 2π-phase modulation in the broadband domain. Here, we demonstrate broadband DOEs utilizing multilayer subwavelength structures with different materials, making it possible to freely control the phase and phase dispersion of the structural units on a much larger scale than monolayer structures. The desired dispersion-control abilities arose due to a dispersion-cooperation mechanism and vertical mode-coupling effects between the top and bottom layers. An infrared design comprised of two vertically concatenated T i O 2 and Si nanoantennas separated by a S i O 2 dielectric spacer layer was demonstrated. It showed an average efficiency of over 70% in the three-octave bandwidth. This work shows enormous value for broadband optical systems with DOEs such as spectral imaging and augmented reality.