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.