  We solve a long-standing set of problems in optics and waves: why does a
volume have only so many useful orthogonal wave channels in or out of it, why
do coupling strengths fall off dramatically past this number, and, indeed, just
what precisely defines that number? Increasingly in applications in
communications, information processing, and sensing, in optics, acoustics, and
electromagnetic waves generally, we need to understand this number. We can
numerically find such channels for many problems, but more fundamentally, these
questions have arguably never had a clear answer or physical explanation. We
have found a simple and general result and intuition that lets us understand
and bound this behavior for any volume. This is based on a tunneling that has
been somewhat hidden in the mathematics of spherical waves: beyond a certain
complexity of the wave, it must tunnel to escape the volume. By counting the
number of waves that do not have to tunnel, we get a simple and precise number
or bound for well-coupled channels, even for arbitrary volumes. The necessary
tunneling for other waves explains the rapid fall-off in their coupling, and
shows all such waves do escape to propagation to some degree after tunneling.
This approach connects multipole expansions in electromagnetic antennas and
nanophotonics smoothly to apparently evanescent waves in large optics. It works
over all size scales, from nanophotonics, small radio-frequency antennas, or
acoustic microphones and loudspeakers up to imaging optics with millions of
channels, and gives a precise diffraction limit for any volume.
