The practice of using complex-valued notation in classical electromagnetic theory is shown to have a more fundamental basis than that of simplifying mathematical manipulations. In the algebraic formulation, the positive and negative time-frequency components of the real-valued field quantities are propagated by two different diffraction operators, each of which is the dual (adjoint) of the other. In addition to the basic distinguishability of these positive and negative time-frequency components, other features distinguish them in general, owing to the constraint of Maxwell’s equations. In the monochromatic case, there are boundary fields for which these other features vanish. For some of these fields, a pseudoscopic image can be propagated. The appearance and location of the pseudoscopic image in holography is shown to be a direct consequence of the unitary property of the diffraction operator in the absence of evanescent waves.
W. DUANE MONTGOMERY, "Basic Duality in the Algebraic Formulation of Electromagnetic Diffraction," J. Opt. Soc. Am. 59, 804-809 (1969)