The use of Hamilton’s mixed and angle characteristic functions in wave and diffraction aberration calculations is theoretically examined. The relation of Hamilton's mixed and angle characteristic functions to a new wave-aberration function is shown. This aberration function is to be used in the Luneburg-Debye diffraction integrals. The mixed and angle characteristic functions as utilized in diffraction theory via the Luneburg-Debye integrals are examined. The mathematical and physical approximations are discussed. The use of the Luneburg-Debye diffraction integrals for image evaluation is examined and some difficulties are pointed out. It is concluded that the above methods should not be used for microwave and radio-frequency imaging systems; they are of limited validity for optical imaging systems.
ARTHUR B. SHAFER, "Hamilton’s Mixed and Angle Characteristic Functions and Diffraction Aberration Theory," J. Opt. Soc. Am. 57, 630-638 (1967)