Increasing demand for high-performance optical devices drives the search for improved fabrication and design paradigms. While photonic circuits have traditionally used structures with two discrete 'binary' height levels, grayscale 'wavy' interfaces, or optical Fourier surfaces, have recently become possible. They provide precise control over the Fourier components that govern the optical response. This capability raises the question: When does a wavy device improve performance and why? Here, we show that wavy integrated processors exhibit superior accuracy and efficiency to binary analogs. Inverse-designed wavy interferometers have the design freedom to minimize outscattering to free space and backreflections. They reach a five-fold lower transmission error than binary counterparts and enable bandwidths up to 300 nm. We formalize these findings for other distinct integrated devices, such as photonic crystals, nanocavities, and beam emitters. A Fourier-optics analysis identifies a trade-off: wavy profiles manipulate light more accurately, whereas binary profiles excel in interaction strength. As such, tailored wavy profiles emit high-quality beams. However, binary profiles remain preferable in photonic crystals and nanocavities, where maximal index contrast is required and binarization-induced higher harmonics are benign. Thus, for quantum information, optical computing, and sensing, optical Fourier surfaces offer a route to miniaturized integrated circuits with improved performance.