Translation:Letter to Arago on biaxial birefringence

by Augustin Jean Fresnel

No. XXXVII.

LETTER OF A. FRESNEL TO F. ARAGO.^[1]

Paris, 21 September 1821.

My dear friend,

Yesterday I finally ascertained that the speed was non-uniform for the ordinary ray in crystals of two axes, as my theoretical ideas on double refraction told me in advance and in such a necessary manner that, if the result of the experiment had not been in conformity with this consequence, I would have been obliged to reject entirely my hypotheses on double refraction; this is easy to see by reading the explanation of the uniformity of the speed of the ordinary ray for crystals of one axis, which I have given in the note inserted in the Annales, where I account for this law by supposing that the elasticity is the same all around the axis.^[2] It is clear that the same thing no longer happens in crystals of two axes, around the line that divides the angle of the two axes into two equal parts, and that the displacements of the molecular rows perpendicular to this line should not develop the same accelerating forces in the plane of the two axes as in a direction perpendicular to that plane; this is also what the experiment confirms.

I glued edge-to-edge two small plates of topaz of the same thickness, cut parallel to this middle line, but of which the one placed on the right had its faces parallel to the plane of the axes, and the one on the left perpendicular to this plane. The face by which they were glued together was a cleavage face and consequently perpendicular to the midline of the two axes. Now I found that the extraordinary rays traversed these two plates with the same speed, in accordance with my theory, while the fringes produced by the ordinary rays were thrown to the left, again as the theory indicated; I have checked several times the direction of polarization of the fringes, so that I am perfectly sure of this result.^[3]

But the difference given to me by observation is smaller than that which I had deduced in advance from the measurements of M. Biot: by my hypothesis, the calculation indicated an interval of 21.1 fringe widths, and the experiment gave me only 16.6; the difference is 5.5 [sic], which is more than a quarter. Would this come from some fault of calculation, or from a notable difference between the optical properties of the limpid topaz that I used and those of the topazes of M. Biot? This is what I do not have time to look for at the moment. But the theoretical ideas that I have adopted on double refraction seem, to me, already confirmed fairly well by this experiment, and especially by their agreement with M. Biot's law on the direction of the planes of polarization and M. Brewster's law on the differences in speed of the ordinary and extraordinary rays.^[4] In truth I have not yet checked the consistency of this second law with my hypothesis except in the particular case where the ray is in the plane of the two virtual axes; but everything leads me to believe that it is supported in the ellipsoid for all other directions whatsoever; this is what I shall check when I have time to do the calculation of it, which seemed a little too long for me to undertake at that moment.

It would be possible that what I have published on the double refraction of crystals of a single axis^[2] gave rise in the mind of M. Young, or of M. Brewster, to the same ideas on that of crystals of two axes. If you should perceive this, I would ask you then to communicate my experiment to the Royal Society of London; in the contrary case, I think it necessary to wait to publish this discovery, until I can present a more complete work on the subject and sufficiently develop my theoretical ideas.^[5]

While you are in England, please try to make an ample collection of observations of the lighthouses and of the views of experienced sailors. Have the English applied gas lighting to any of their fixed-flame lighthouses?^[6]

....^[7]

Farewell, my dear friend.

Signed:  A. Fresnel.

Notes

1. Translated from A. Fresnel (ed. H. de Sénarmont, E. Verdet, and L. Fresnel), Oeuvres complètes d'Augustin Fresnel (hereinafter cited as OC), vol. 2 (1868), pp. 257–9.  (Cf. pp. 257, 258, and 259 at French Wikisource.)
2. A. Fresnel, "Note sur le calcul des teintes que la polarisation développe dans les lames cristallisées" et seq., Annales de Chimie et de Physique, Ser. 2, vol. 17, pp. 102–11 (May 1821), 167–96 (June 1821), 312–15 ("Postscript", July 1821), reprinted (with added section nos.) in OC, vol. 1 (1866), pp. 609–48, translated as "On the calculation of the tints that polarization develops in crystalline plates, & postscript", doi:10.5281/zenodo.4058004, 2021, at § 14.
3. In a biaxial birefringent crystal, there are three mutually perpendicular planes—one through the axes, one through the "midline" of the acute angle between the axes, and one perpendicular to that midline—in each of which the speed of light is independent of direction within the plane provided that the polarization has the electric displacement vector (Fresnel's "displacement" or vibration) perpendicular to that plane. In the plane perpendicular to the midline, the rays with this polarization are designated by Fresnel as the "extraordinary" rays; this terminology takes the midline between the axes as analogous to the axis of a uniaxial crystal.
4. These laws are also known respectively as Biot's dihedral law and Biot's sine law;  cf. A. Fresnel, "Second Memoir" on double refraction (1827), translated by A.W. Hobson in R. Taylor (ed.), Scientific Memoirs, vol. V, London: Taylor & Francis, 1852, pp. 238–333, esp. pp. 312–17 (sine law), 320–23 (dihedral law).
5. This took just over six months; see OC, vol. 2, pp. 261–442.
6. An administrative decision of 19 June 1819, prompted by Arago, had called the engineer of bridges and roads, A. Fresnel, to contribute to the experiments proposed by the Lighthouse Commission for the improvement of the lighting of our coasts. This appointment resulted in the invention and creation of the system of lenticular lighthouses used today by all maritime powers. From the beginning of 1820, Fresnel had caused M. Soleil Sr. to manufacture a prototype large polygonal lens intended to form one of the panels of a rotating dioptric drum. To illuminate these effectively, however, it was necessary to maintain a sufficiently voluminous and intense flame at their common focus — a problem that the inventor had managed to solve, in collaboration with Arago, by means of a nozzle with concentric wicks fed with oil by a clockwork mechanism; but the constraints and the risks of disturbance inherent in this type of lamp made Fresnel think of resorting to the use of gas.  [Note by Léonor Fresnel in the OC, vol. 2 (1868), p. 259. Other notes are by the Translator.]
7. Redacted by the editors of the OC.