Judiciously distributing laser emitters to shape the desired far field patterns

The far-field pattern of a simple one-dimensional laser array of emitters radiating into free space is considered. In the path of investigating the inverse problem for their near fields leading to a target beam form, surprisingly we found that the result is successful when the matrix of the corresponding linear system is not well-scaled. The essence of our numerical observations is captured by an elegant inequality defining the functional range of the optical distance between two neighboring emitters. Our finding can restrict substantially the parametric space of integrated photonic systems and simplify significantly the subsequent optimizations.