Fresnel's original interpretation of complex numbers in 19th century optics Available to Purchase

This was a cumbersome process where different physical principles were invoked in a constant interplay with experimental results. For a detailed account of how Fresnel came up with the transverse wave hypothesis, see Chapter 8 in Ref. 7, and for the different assumptions leading to Fresnel's equations, see Appendix 17 in the same reference.

If the media have the same elasticity, from general considerations about the propagation of disturbances in elastic media, namely, the influence of the medium's elasticity (E) and density (ρ) on the velocity of propagation (⁠$v=E/ρ$⁠), Fresnel was able to find the relationship $v12ρ1=v22ρ2$⁠, which combined with Snell's law yields $ρ1/ρ2=sin2 θt/ sin2 θi$⁠. The ratio between the densities is needed when conservation of vis viva (kinetic energy) is applied in the derivation of Fresnel's equations. Details are found both in Fresnel's original (Ref. 5, p. 771) as well as in Appendix 17 of Ref. 7.

Nowadays these equations are derived by treating light as an electromagnetic wave with Maxwell's equations and applying boundary conditions to the components of the $E→$ and $H→$ fields. See, for instance, Section 1.5.2 in Ref. 9 or Chapter 4 in Ref. 13.

The interested reader will notice some differences concerning signs when comparing how Eqs. (1) and (2) are presented in different parts of Ref. 5. This is due to a convention issue which is carefully explained by Fresnel in item 18 (Ref. 5, pp. 787–790). The version presented here is in agreement with this convention (see Ref. 5, p. 789) and is also the modern version that can be found, for instance, in Ref. 9.

The French term in the original is périodes de vibration. It is not to be confused with our modern understanding of period (inverse of the frequency), but it represents what we denote nowadays by phase.

One physical way to understand the reason for the phase shift is to consider that the total reflection is not happening completely/abruptly at the interface, but that an evanescent wave does penetrate the second medium. Thus, it gains some phase shift when it is reflected back to the incident medium. This also explains the so-called frustrated total internal reflection (see Ref. 17).

The relationship between the phase difference of the orthogonal components and the kind of polarization is quite hard to visualize. Animations like the one available at http://emanim.szialab.org/ can be very helpful.

One can also treat Eq. (8) as a function of the angle of incidence and search for its extremes. Fresnel does this in Ref. 5 (p. 791) and shows that for Saint-Gobain glass (n = 1.51) the maximum phase shift that can be obtained with only one total internal reflection is $45°56′12$ corresponding to an angle of incidence of $51°20′13$⁠.