Upper bound on the diffraction efficiency of phase-only fanout elements

For one-dimensional binary-phase [(0, pi) and (0, non-pi)] fanout elements and for one-dimensional continuous or multilevel quantized phase fanout elements, an upper bound on diffraction efficiency is presented for fanouts ranging from 2 to 25. The upper bound is determined by optimizing with respect to the array phase the upper bound on diffraction efficiency for a coherent array. To determine the upper bound for binary-phase gratings, restrictions on the array phase are imposed. For fanouts that are >5, the upper bound on the diffraction efficiency for continuous phase fanouts ranges between 97 and 98%; for (0, pi)-binary-phase fanouts the upper bound ranges between 83 and 84%; and for (0, non-pi)-binary-phase, between 87 and 88%.