To make high quality long-range periodic nanostructures in a transparent or semi-transparent substrate, the transparent or semi-transparent substrate is scanned with a linearly polarized laser beam generated by a femtosecond laser and exceeding a predetermined energy/pulse threshold along a scanning path. Sub-diffraction limit structures are formed as periodic planes of modified material in the transparent or semi-transparent substrate extending along the scanning path. The modified material can then be chemically etched to form cavities.