Lenses and Waves

1677-1679 –WAVES OF LIGHT 177
He began with a qualitative
account of spheroidal waves
produced by a perpendicularly
incident wave (Figure 67). 50 RC is
part of a plane wave incident
perpendicularly on the surface AB
of the crystal, so that all points
RHhhC arrive in AKkkB at the
same time. Suppose spheroidal
wavelets SVT spread around these
points, as the speed of
propagation in the direction AV is
larger than in the direction AZ.
According to Huygens’ principle
the common tangent of these spheroidal waves is the refracted wave.
“And it is thus that I have comprehended, what had seemed to me very difficult, how a
ray perpendicular to a surface could suffer refraction on entering the transparent body;
seeing that the wave RC, having come to the aperture AB, continued forward thence,
extending between the parallels AN, BQ yet itself remaining always parallel to AB, such
that here the light does not extend along lines perpendicular to its waves, as in ordinary
refraction, but these lines cut the waves obliquely.” 51
Figure 67 Refraction of the perpendicular.
Thus the application of Huygens’ principle applied to spheroidal waves
showed that these could explain the refracted perpendicular. Huygens had
solved the original problem of strange refraction, as the refracted
perpendicular implied that waves would not be at right angles to the line of
their extension. Strange refraction was strange precisely because of this: the
propagation of light in Iceland crystal is extraordinary and produces waves
that are not normal to rays.
At this point in the Traité de la Lumière the problem was solved, but only
in principle. For Huygens the most important question still had to be
answered: how spheroidal waves could explain strange refraction. The answer,
the exact properties of the spheroid, was that of 6 August 1677.
In Traité de la Lumière, he began with observing that in the principal
sections of each face of the crystal (the dotted lines in Figure 68) strange
refraction behaves the same. Consequently, the spheroid must have the same
section in all three planes. This is so when the axis of the spheroid is also the
axis of the obtuse solid angle C of the crystal. Choosing plane GCF, Huygens
drew the ellipse PSG, the cross-section of the spheroid around center C
(Figure 69). CS is the axis of the obtuse solid angle C as well as the axis of
50 Traité, 60-62.
51 Traité, 61-62. “Et c’est ainsi que j’ay compris, ce qui m’avoit paru fort difficile, comment un rayon
perpendiculaire à une surface pouvoit souffrir refraction en entrant dans le corps transparent; voyant que
l’onde RC, estant venue à l’ouverture AB, continuoit de là en avant à s’étendre entre les paralleles AN, BQ
demeurant pourtant elle mesme tousiours parallele à AB, de sorte qu’icy la lumiere ne s’étend pas par des
lignes perpendiculaires à ses ondes, comme dans la refraction ordinaire, mais ces lignes coupent les ondes
obliquement.”