Controlling the flux of photons is crucial in many areas of science and technology. Artificial materials with nano-scale modulation of the refractive index, such as photonic crystals, are able to exercise such control and have opened exciting new possibilities for light manipulation. An interesting alternative to such periodic structures is the class of materials known as quasi-crystals, which offer unique advantages such as richer Fourier spectra. Here we introduce a novel approach for designing such richer Fourier spectra, by using a periodic structure that allows us to control its Fourier components almost at will. Our approach is based on binary gratings, which makes the structures easy to replicate and to tailor towards specific applications. As an example, we show how these structures can be employed to achieve highly efficient broad-band light trapping in thin films that approach the theoretical (Lambertian) limit, a problem of crucial importance for photovoltaics.