On Conical Refraction

By the Rev. H. LLOYD, Professor of Natural and Experimental Philosophy in the University of Dublin.

[Report of the Third Meeting of the British Association for the Advancement of Science held at Cambridge in 1833 (John Murray, London, 1834), 370--373.]

Professor Lloyd gave a brief account of the experiments by which he established the existence of conical refraction in biaxal crystals, in conformity with the theoretical anticipations of Professor Hamilton.

The substance employed in these experiments was arragonite, which was selected chiefly on account of the magnitude of its biaxal energy. The specimen was one of remarkable purity, procured by Mr. Dollond. Its thickness was ·49 of an inch, and its parallel faces were perpendicular to the line bisecting the optic axes, being cleavage planes of the crystal.

The first case of conical refraction examined by the author was that called by Professor Hamilton external conical refraction. It was expected to take place when a single ray passes within the crystal in the direction of the line connecting two opposite cusps on the wave-surface. When this is the case, Professor Hamilton has shown that there should be a cone of rays without, the magnitude of which will depend on the biaxal energy of the crystal. In the case of arragonite, the angle of the cone, calculated from the elements of the crystal as determined by M. Rudberg, amounts to 3° very nearly.

A thin metallic plate, perforated with a minute aperture, was placed on each face of the crystal, and these were adjusted in such a manner that the line connecting the apertures should coincide nearly with one of the optic axes. The flame of a lamp was then brought near the first surface of the crystal, and in such a position that the central part of the beam converging to the aperture should have an incidence of between 15° and 16°. When the adjustment was completed, there appeared, on looking through the aperture on the second surface, a brilliant luminous circle with a small dark space around its centre; and in this central disk were two bright points, separated by a well-defined dark line. When the plate on the second surface was slightly shifted, so that the line connecting the two apertures no longer coincided accurately with the line joining the cusps on the wave-surface, the phaenomena rapidly changed, and ultimately resolved themselves into the two pencils into which a single ray is divided under ordinary circumstances.

The incident converging cone was sometimes formed by a lens of short focus, placed at the distance of its focal length from the surface. In this case the lamp was removed to a considerable distance, and the plate on the first surface dispensed with. The same experiment was repeated with the sun's light, instead of that of a lamp, and the emergent cone of rays thus formed was of sufficient intensity to be reflected from a screen at some distance. In this manner the section of the cone was observed at various distances from its summit.

When these phaenomena were examined in detail, and compared with the results of theory, they appeared to differ in two important particulars. In the first place, the observed cone was very nearly a solid cone of rays, while that of theory was but a conical surface. Secondly the two cones differed widely in magnitude, the angle of the experimental cone being nearly double that of the theoretical one. This discordance between the results of experiment and those of theory, the author conceived to arise from the rays which were inclined to the optic axis at small angles, and which were transmitted through the aperture on the second surface in consequence of its sensible magnitude. To examine this point he proceeded in the next place to try the effects of apertures of various dimensions. The effects of these variations in the resulting phaenomena corresponded exactly to his preconceived views. The rays which in the first experiments filled the whole of the conical space, parted in the centre, when the aperture was much diminished; and the section of the cone, instead of a complete luminous circle was reduced to a luminous annulus, whose breadth diminished with the aperture. Simple theoretical considerations showed that the angle of the true cone in this case must be, very nearly, half the sum of the angles of the observed interior and exterior cones; and when this correction was applied to the measurements, the resulting angle agreed, as nearly as could be expected, with that deduced from theory.

A remarkable variation of the phaenomena was obtained by substituting a narrow linear aperture for the small circular one, in the plate next the lamp in the first-mentioned mode of performing the experiment; and by adjusting it so that the plane passing through it and the aperture on the second surface should coincide with the plane of the optic axes. In this case, according to the received theory, all the rays transmitted through the two apertures should be refracted doubly in the plane of the optic axes, so that no part of the line should appear enlarged in breadth on looking through the aperture on the second surface. But if Professor Hamilton's conclusion be physically exact, the ray which proceeds in the direction of the line joining two opposite cusps on the wave-surface should be refracted in every plane. This was accordingly found to be the case. In the neighbourhood of each of the optic axes the luminous line swelled out on either side of the plane of the axes in an oval curve. This curve is the conchoid of Nicomedes, whose asymptot is the line on the first surface; and its variations of form, as the plane passing through the two apertures deviated from the plane of the optic axes, were highly curious and remarkable.

Examining the state of polarization of the rays composing the emergent cone, Mr. Lloyd discovered that they observed the following law, namely, that ``the angle between the planes of polarization of any two rays of the cone is half the angle contained by the planes passing through the rays themselves and its axis.'' This law, it is easy to show, is in perfect accordance with theory.

Internal conical refraction should take place, according to Professor Hamilton, when a single ray has been incident externally upon a biaxal crystal in such a manner that one of the refracted rays may coincide with an optic axis. The ray, in this case, ought to be divided into a cone of rays within the crystal, the angle of which in the case of arragonite is 1° 55'. The rays forming the cone will be refracted at the second surface of the crystal in directions parallel to the ray incident on the first; so that they will form a small cylinder of rays in air, whose base is the section of the cone formed by the surface of emergence.

The minuteness of this phaenomenon, and the perfect accuracy required in the incidence, render it much more difficult of detection than the former. A very fine ray of light proceeding from a distant lamp was suffered to fall upon the crystal, and the position of the latter altered with extreme slowness, so as to change the incidence very gradually. When the required position was attained, the two rays suddenly spread into a continuous circle, whose diameter was apparently equal to their former distance. The same experiment was repeated with the sun's light, and the emergent cylinder received on a small screen of paper at various distances from the crystal. No sensible enlargement of the section was visible on increasing the distance.

The magnitude of the angle of the cone of rays within the crystal was ascertained experimentally, and agreed within very narrow limits with that deduced from theory. The rays composing the cone were all polarized in different planes, and the law connecting the planes of polarization was the same as that already found in the case of external conical refraction.

A full account of these experiments is given in a memoir read before the Royal Irish Academy, on the 28th January 1833, and ordered to be published in the 17th volume of its Transactions.


D.R. Wilkins
School of Mathematics
Trinity College, Dublin