  We analyze how complicated a linear optical component has to be if it is to
perform one of a range of functions. Specifically, we devise an approach to
evaluating the number of real parameters that must be specified in the device
design or fabrication, based on the singular value decomposition of the linear
operator that describes the device. This approach can be used for essentially
any linear device, including space-, frequency-, or time-dependent systems, in
optics or in other linear wave problems. We analyze examples including spatial
mode converters and various classes of wavelength demultiplexers. We consider
limits on the functions that can be performed by simple optical devices such as
thin lenses, mirrors, gratings, modulators, and fixed optical filters, and
discuss the potential for greater functionalities using modern nanophotonics.
