Optical Fourier surfaces

Gratings¹ and holograms² use patterned surfaces to tailor optical signals by diffraction. Despite their long history, variants with remarkable functionalities continue to be developed^3,4. Further advances could exploit Fourier optics⁵, 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^6-8 and templating⁹ 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 engineering¹⁰. Consequently, we overcome a previous limitation in photonics by creating an ultrathin 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^11,12, quasicrystals^13,14 and holograms^15,16, demonstrating a variety of previously unattainable diffractive surfaces. This approach may find application in optical devices (biosensors¹⁷, lasers^18,19, metasurfaces⁴ and modulators²⁰) and emerging areas in photonics (topological structures²¹, transformation optics²² and valleytronics²³).