Optical Fourier surfaces
Gratings and holograms are patterned surfaces that tailor optical signals by diffraction. Despite their long history, variants with remarkable functionalities continue to be discovered. Further advances could exploit Fourier optics, which specifies the surface pattern that generates a desired diffracted output through its Fourier transform. To shape the optical wavefront, the ideal surface profile should contain a precise sum of sinusoidal waves, each with a well-defined amplitude, spatial frequency, and phase. However, because fabrication techniques typically yield profiles with at most a few depth levels, complex 'wavy' surfaces cannot be obtained, limiting the straightforward mathematical design and implementation of sophisticated diffractive optics. Here we present a simple yet powerful approach to eliminate this design-fabrication mismatch by demonstrating optical surfaces that contain an arbitrary number of specified sinusoids. We combine thermal scanning-probe lithography and templating to create periodic and aperiodic surface patterns with continuous depth control and sub-wavelength spatial resolution. Multicomponent linear gratings allow precise manipulation of electromagnetic signals through Fourier-spectrum engineering. Consequently, we immediately resolve an important problem in photonics by creating a single-layer grating that simultaneously couples red, green, and blue light at the same angle of incidence. More broadly, we analytically design and accurately replicate intricate two-dimensional moiré patterns, quasicrystals, and holograms, demonstrating a variety of previously impossible diffractive surfaces. Therefore, this approach provides instant benefit for optical devices (biosensors, lasers, metasurfaces, and modulators) and emerging topics in photonics (topological structures, transformation optics, and valleytronics).