Conical refraction

For external conical refraction, light is focused at a single point aperture on the slab of biaxial crystal, and exits the slab at the other side at an exit point aperture.

This effect was predicted in 1832 by William Rowan Hamilton[1] and subsequently observed by Humphrey Lloyd in the next year.

[2] It was possibly the first example of a phenomenon predicted by mathematical reasoning and later confirmed by experiment.

[3] The phenomenon of double refraction was discovered in the Iceland spar (calcite), by Erasmus Bartholin in 1669. was initially explained by Christiaan Huygens using a wave theory of light.

In 1813, David Brewster discovered that topaz has two axes of no double refraction, and subsequently others, such as aragonite, borax and mica, were identified as biaxial.

[4] At the same period, Augustin-Jean Fresnel developed a more comprehensive theory that could describe double refraction in both uniaxial and biaxial crystals.

Fresnel had already derived the equation for the wavevector surface in 1823, and André-Marie Ampère rederived it in 1828.

[5] Many others investigated the wavevector surface of the biaxial crystal, but they all missed its physical implications.

[4] William Rowan Hamilton, in his work on Hamiltonian optics, discovered the wavevector surface has four conoidal points and four tangent conics.

Lloyd observed external conical refraction 14 December with a specimen of arragonite from the Dollonds, which he published in February.

At the same time, Hamilton also exchanged letters with George Biddell Airy.

Airy had independently discovered that the two sheets touch at conoidal points (rather than tangent), but he was skeptical that this would have experimental consequences.

The rays of the internal cone emerged, as they ought, in a cylinder from the second face of the crystal; and the size of this nearly circular cylinder, though small, was decidedly perceptible, so that with solar light it threw on silver paper a little luminous ring, which seemed to remain the same at different distances of the paper from the arragonite.

[10] In 1833, James MacCullagh claimed that it is a special case of a theorem he published in 1830[11] that he did not explicate, since it was not relevant to that particular paper.

The indices are the major and minor axes of the ellipse of intersection between the plane and the index ellipsoid.

At precisely 4 directions, the intersection is a circle (those are the axes where double refraction disappears, as discovered by Brewster, thus earning them the name of "biaxial"), and the two sheets of the surface of wavevectors collide at a conoidal point.

To be more precise, the surface of wavevectors satisfy the following degree-4 equation (,[9] page 346):

The major and minor axes are the solutions to the constraint optimization problem:

, we obtain the local geometry of the surface, which is a cone subtended by a circle.

Further, there exists 4 planes, each of which is tangent to the surface at an entire circle (a trope conic, as defined later).

The special tangent plane to the surface touches it at two points that make an angle of

In this case, the conoidal point is approximately at the center of the tangent circle surrounding it, and thus, the cone of light (in both the internal and the external refraction cases) is approximately a circular cone.

In the case of external conical refraction, we have one ray splitting into a cone of planar waves, each corresponding to a point on the tangent circle of the wavevector surface.

[23] The classical theory of conical refraction was essentially in the style of geometric optics, and ignores the wave nature of light.

[13][24][3] The angle of the cone depends on the properties of the crystal, specifically the differences between its principal refractive indices.

The effect is typically small, requiring careful experimental setup to observe.

Poggendorff observed two rings separated by a thin dark band.

Potter observed in 1841[27] certain diffraction phenomena that were inexplicable with Hamilton's theory.

[28] The study of conical refraction has continued since its discovery, with researchers exploring its various aspects and implications.

Some recent work includes: Conical refraction was also observed in transverse sound waves in quartz.