How to design any linear optical component and how to avoid it

Though we can readily understand any particular linear optical device, we have not in general known how to design one to do just what we want. At worst, we have to resort to blind, trial-and-error iterative processes, with no guarantee that the device is even possible. For example, current challenging practical devices include efficient arbitrary spatial mode splitters and converters for communications through free-space or in multimode fibers, especially avoiding any fundamental power splitting loss. Here we show a new constructive, progressive, non-iterative method to design any arbitrary linear optical device or, indeed, any linear operation on waves, including microwaves, acoustics and quantum mechanical superpositions. We propose practical approaches for spatial optical devices that could be implemented, for example, in silicon photonics. The existence of this design approach also proves, in the spirit of a universal machine, that any linear wave device is possible in principle. Surprisingly, we show that all these designs can be completed without performing any calculations at all.