A method of creating thin wafers of single crystal silicon, sapphire and similar materials, wherein an ingot of single crystalline material, or a ribbon of single crystalline material is cleaved, in a plane parallel to a surface, with laser light focused to a line in the desired plane of cleavage, near the growing cleavage furrow. The light is of a wavelength that the material is transparent to, but for which the material has strong two- or three-photon absorption. Consequently the light is not appreciably absorbed until it reached the desired focal line. The light is presented in an extremely short pulse, which heats and expands the material at the line focus, before the heat can be dissipated. This expansion creates tangential stresses around the focal line. These stresses are designed to be precisely normal to the growing cleavage furrow. Therefore the stresses are able to induce cleavage in the desired plane, without inducing cleavage in other possible cleavage planes that may happen to intersect with the growing cleavage edge. In this way, extremely thin wafers and ribbon shaped wafers can be produced, with extremely high quality cleaved faces. Methods of initiating the cleavage furrow and separating the cleaved wafer from the rest of the crystal are also discussed.