Orthogonality of non-coaxial Laguerre-Gaussian beams

The conventional orthogonality of Laguerre-Gaussian (LG) beams is widely recognized and serves as the fundamental basis for mode division multiplexing utilizing LG beams as channels. However, this conventional orthogonality holds true only when two LG beams share the same axis. Whether non-coaxial LG beams, with parallel axes separated by a distance, are orthogonal to each other remains an open question. To address this issue and reveal the orthogonality of non-coaxial LG beams, we analytically derive the projection of one LG beam onto another. This projection exhibits a helical phase and a rotationally symmetric amplitude with dark rings present. These dark rings indicate that two non-coaxial LG beams are indeed orthogonal to each other when their axes are appropriately separated. Furthermore, any pair of non-coaxial LG beams within a set also exhibit orthogonality when their axes are fixed at a certain separation distance. This novel form of orthogonality predicts an innovative compound multiplexing technique that simultaneously combines mode and space, enabling more efficient packing of multiple LG beams within a single transmission aperture while maintaining parallel and closely spaced beam axes.