  Modes generally provide an economical description of waves, reducing
complicated wave functions to finite numbers of mode amplitudes, as in
propagating fiber modes and ideal laser beams. But finding a corresponding mode
description for counting the best orthogonal channels for communicating between
surfaces or volumes, or for optimally describing the inputs and outputs of a
complicated optical system or wave scatterer, requires a different approach.
The singular-value decomposition approach we describe here gives the necessary
optimal source and receiver "communication modes" pairs and device or scatterer
input and output "mode-converter basis function" pairs. These define the best
communication or input/output channels, allowing precise counting and
straightforward calculations. Here we introduce all the mathematics and physics
of this approach, which works for acoustic, radio-frequency and optical waves,
including full vector electromagnetic behavior, and is valid from nanophotonic
scales to large systems. We show several general behaviors of communications
modes, including various heuristic results. We also establish a new "M-gauge"
for electromagnetism that clarifies the number of vector wave channels and
allows a simple and general quantization. This approach also gives a new modal
"M-coefficient" version of Einstein's A&B coefficient argument and revised
versions of Kirchhoff's radiation laws. The article is written in a tutorial
style to introduce the approach and its consequences.
