It is not known exactly when Fresnel made this last step, because there is no relevant documentation from 1820 or early 1821 (perhaps because he was too busy working on lighthouse-lens prototypes; see below). But he first published the idea in a paper on " Calcul des teintes… " ("calculation of the tints…"), serialized in Arago's Annales for May, June, and July 1821. In the first installment, Fresnel described "direct" (unpolarized) light as "the rapid succession of systems of waves polarized in all directions", and gave what is essentially the modern explanation of chromatic polarization, albeit in terms of the analogy between polarization and the resolution of forces in a plane, mentioning transverse waves only in a footnote. The introduction of transverse waves into the main argument was delayed to the second installment, in which he revealed the suspicion that he and Ampère had harbored since 1816, and the difficulty it raised. He continued: