1Introduction 2Organization of This Paper 3Introduction to SVD and Waves—Sets of Point Sources and Receivers 4Introductory Example—Three Sources and Three Receivers 5Scalar Wave Examples with Point Sources and Receivers 6Mathematics of Continuous Functions, Operators, and Vector Spaces 7Communications Modes and Common Families of Functions 8Extending to Electromagnetic Waves 9Quantizing the Electromagnetic Field Using the M-Gauge 10Linear Scatterers and Optical Devices 11Mode-Converter Basis Sets as Fundamental Optical Descriptions 12Conclusions

(1) Input There are zero or more quantities that are externally supplied. (2) Output At least one quantity is produced. (3) Definiteness Each instruction is clear and unambiguous. (4) Finiteness If we trace out the instructions of an algorithm, then for all cases, the algorithm terminates after finite number of steps. (5) Effectiveness Every instruction must be basic enough to be carried out, in principle, by a person using only pencil and paper. It is not enough that each operation be definite as in(3); it also must be feasible