In 1950, Giuliano Toraldo di Francia published a paper [1] with the title "Parageometrical Optics" followed by several other papers on the subject, I dare say culminating in a beautiful design work of a microwave antenna [2]. Toraldo also presented his ideas in a rather brief chapter in his otherwise excellent book "Onde Elettromagnetiche" published in 1953. His work on parageometric optics seems to have preceded or at least be simultaneous with Keller's Geometric Theory of Diffraction.

Parageometric optics is supposed to be an extension of conventional geometric optics to include diffraction effects. In his analysis, Toraldo shows that the various diffractive orders resulting from an arbitrary (amplitude/phase) transparency illuminated by a plane wave will follow Malus' law for the associated ray systems having orthogonal surfaces and thus the propagation of diffraction orders can be treated individually and "geometrically" and, furthermore, the transparency can be of arbitrary (smooth) shape.

It seems to me that since 1960 or so nothing has been written on the subject, and by now is completely forgotten despite its eminent intuitive appeal; Born&Wolf does not mention it at all, not even as a pedagogic device. My question is why? Is anything fundamentally so wrong with it that the idea is not worth discussing?

[1]:"Parageometrical Optics", JOURNAL OF THE OPTICAL SOCIETY OF AMERICA, 1950 vol.40, 9

[2]: "An Application of Parageometrical Optics to the Design of a Microwave Mirror", IRE Trans. Antennas and Propagation, January 1958

I asked this on https://physics.stackexchange.com/questions/633842/diffraction-physics-v-parageometric-optics-whatever-happened-to-it but got no reply.


1 Answer 1


There was nothing wrong with parageometrical optics mathematically, it just did not take. It happens. Compare Google ngram for parageometrical to the ngram for nomography. And nomography had everything, mathematical depth, intuitive appeal, wide use by engineers for computations and visualization, and yet we see the same decline of interest since 1960s. Why? Because computers. With all the tools they provide there just wasn't much demand for clever geometric shortcuts like nomography or parageometrical optics. See e.g. Mínguez-Vega et al. (2007) using wave optics, complete with numerical simulations, to redo the "parageometrical" design of Amako et al. (2002) for a diffractive-refractive lens triplet (they do not even mention di Francia).

Another possible reason is that it did not generate enough new insights. One can handle diffraction by the more conventional Fresnel approach, including explanations of Ronchi-di Francia's antennas, see Cornbleet, Microwave Optics, p.81 doing it in 1973 after mentioning the original parageometrical design in passing. By then, di Francia himself moved on to greener pastures, see Geodesic Lenses for Guided Optical Waves co-authored by him. As very similar ngrams for horn angles and catastrophe theory show, they had their heyday in the 1940s and 1980s, respectively, and then fizzled out much like parageometrical optics. The reason was again the scarcity of new insights. It is not enough for an idea to be neat to take, it has to be sufficiently fruitful.

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    $\begingroup$ Your analogy with nomograms is very interesting, because whatever little intuitive understanding I have in this subject has started with me by having stumbled on Toraldo's name in the context image information and resolving power, and read his antenna design that I found so clever and looked for more. I thought it was incredibly clever, clear, intuitive, etc., and of course, so are nomograms! I still do not understand that neither is not used, not even as a teaching tool. Thank you. $\endgroup$
    – hyportnex
    Jan 17, 2022 at 14:08

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