Replication of band structure in an arbitrary wave vector by holographic modulation

Diffraction is a fundamental physical phenomenon that occurs when scattering wave sources are arranged periodically. For example, when x rays are incident on the common salt crystals, diffraction occurs owing to the periodic structure of the chlorine and sodium atoms. Recent advances in nanotechnology have enabled artificial control of the scattering points even in visible wavelengths. These artificial structures are called photonic crystals and are related to many interesting phenomena. Dispersion relations between wave vector and frequency are called band structures. Similar to the band structures of electrons and holes in crystals, photonic bands for dispersion determine the direction of propagation of light waves by diffraction of the periodic structures. Herein, we introduce band engineering which enables multiple replications of bands in arbitrary wave vectors by holographic modulation. We have already realized a patterned light source named ``iPMSEL'' as a typical application of holographically engineered photonic band. The iPMSEL has been realized by combining photonic-crystal surface-emitting lasers and holographic engineering. It can emit arbitrary multiple beam patterns and even photographic images from a tiny semiconductor chip without any optical components. In this paper, we study the iPMSEL from the viewpoint of the photonic-band replication and operation mechanism. While this work is based on photonics, the findings can also be applied to other fields of physics concerned with diffraction. We believe that this work provides useful methods with respect to periodic structures within all scale ranges.