Inverse symmetric Dammann gratings

Inverse symmetric Dammann grating is a special grating, whose transition points are reflection symmetric about the midpoint with inverse phase offset in one period. It can produce even-numbered or odd-numbered array illumination when the phase modulations are π or a specific value. Numerical solutions optimized by the steepest-descent algorithm for binary phase and multilevel phases with splitting ratio from 1×4 to 1×14 are given. Fabrication of 1×6 array without the zero-order intensity and 1×7 array with the zero-order intensity are made from the same amplitude mask. A 6×6 output without the crossed zero-orders was achieved by crossing two one-dimensional 1×6 inverse symmetric Dammann gratings. This grating may have potential value for practical applications.