According to Fourier optics, the surface profile of an ideal diffraction
grating should contain a precise sum of sinusoidal waves. However,
because fabrication techniques typically yield profiles with only two depth
levels, complex “wavy” surfaces cannot be obtained, limiting the
straightforward design and implementation of sophisticated diffractive
surfaces. Here, we eliminate this design–fabrication mismatch and
produce optical surfaces with an arbitrary number of specified sinusoids,
yielding previously unattainable diffractive surfaces including intricate
two-dimensional moiré patterns, quasicrystals, and holograms. We then
show that such patterns can be reduced to nanometer length scales,
creating wavy Fourier surfaces for 2D electronics.