In 1823, Fresnel published an original (physical) interpretation of complex numbers in his investigations of refraction and reflection of polarized light. This is arguably the first time that complex numbers were given a physical interpretation, which led to a better understanding of elliptical and circular polarizations. This rather unknown episode of the history of physics is described in this work, and some of the pedagogical lessons that can be extracted from it are discussed.

1.
Paul J.
Nahin
, Imaginary Tale: The Story of 1 (
Princeton U.P.
,
Princeton
,
1998
).
2.
Kirsti
Andersen
, “
Wessel's work on Complex Numbers and its Place in History
,” in
Caspar Wessel. On the Analytical Representation of Direction. An Attempt Applied Chiefly to Solving Plane and Spherical Polygons
, edited by
B.
Branner
and
J.
Lützen
(
The Royal Danish Academy of Science and Letters
,
Copenhagen
,
1999
), pp.
65
98
.
3.
Salomon
Bochner
, “
The significance of some basic mathematical conceptions for physics
,”
Isis
54
(
2
),
179
205
(
1963
).
4.
Umberto
Bottazzini
and
Jeremy
Gray
,
Hidden Harmony–Geometric Fantasies: The Rise of Complex Function Theory
(
Springer-Verlag
,
New York
,
2013
).
5.
Augustin-Jean
Fresnel
, “
Mémoire sur la loi des modifications que la réflexion imprime à la lumière polarisée
,” Read in 7 January 1823. In Oeuvres, 1: 767–799.
6.
Augustin-Jean
Fresnel
, “
Extrait d'un Mémoire sur la loi des modifications imprime à la lumière polarisée par sa réflexion totale dans l'intérieur des corps transparents
,” Bulletin de la Société Philomathique (1823). In Oeuvres, 1: 753–762.
7.
Jed Z.
Buchwald
,
Rise of the Wave Theory of Light: Optical Theory and Experiment in the Early Nineteenth Century
(
The University of Chicago Press
,
Chicago
,
1989
).
8.
Edwin T.
Whittaker
,
History of the Theories of Aether and Electricity: From the Age of Descartes to the Close of the Nineteenth Century
(
Longmans, Green, and CO
,
London
,
1910
).
9.
Max
Born
and
Emil
Wolf
,
Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light
, 6th ed. (
Pergamon
,
Exeter
,
1980
).
10.

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.

11.

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=sin2θt/sin2θi. 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.

12.

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.

13.
Eugene
Hecht
,
Optics
, 4th ed. (
Addison-Wesley Longman, Inc.
,
New York
,
2002
).
14.

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.

15.

The French term in the original is périodesdevibration. 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.

16.

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).

17.
S.
Zhu
,
A. W.
Yu
,
D.
Hawley
, and
R.
Roy
, “
Frustrated total internal reflection: A demonstration and review
,”
Am. J. Phys.
54
(
7
),
601
607
(
1986
).
18.

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.

19.

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°5612 corresponding to an angle of incidence of 51°2013.

20.
Augustin-Jean
Fresnel
, Note sur la polarisation circulaire (1823). In Oeuvres, 1: 763–766.
21.
Richard P.
Feynman
,
R. B.
Leighton
, and
M.
Sands
,
The Feynman Lectures on Physics
, Vol. 1 (
Addison-Wesley
,
Reading MA
,
1965
).
22.
G.
Horváth
and
D.
Varjú
, “
Circularly polarized light in nature
” in
Polarized Light in Animal Vision
(
Springer
,
Berlin, Heidelberg
,
2004
).
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