# Internal conical refraction

The photon beam passage through a CRE can be divided into two successive phases: the beam in CRE and the beam after CRE. The case when the beam before CRE is describable by scalar or vector field is considered as well elucidated in the frame of the classical theory — the phenomenon of internal conical refraction.

The case of beam in CRE is exactly solved for the Poynting vector (Feodorov and Filipov, 1967 or earlier). The key moments are:

1) For one period of the vibrations of the incident wave, the Poynting vector makes two full revolutions along conical surface. If the beam frequency is, the Poynting vector turns with frequency .

2) The trajectories of the energy movement are straight lines. Their directions depend on the wave phase at the input base of CRE. When the phase changes from to , the family of lines covers the whole conical surface.

The beam after is a transformation of the beam before. This transformation is found by Belsky and Hapalyuk (1978) for optical axis without optical activity, and in the general case by Belsky and Stepanov (2002); shortly BHS-transformation. The BHS-transformation gives semiclassical upgrade with consequences leading far away from the classical theory.

In the ab initio calculations of the BHS-transformation the beam before is considered to be two-component vector field (vector solution of the paraxial wave equation) with one peculiar plane. There is not radial phase variation on this peculiar plane. This plane is the isomomentum plane of the vector beam.

The special case of scalar beam is obtained as randomly polarized beam. For this special case the beam after evolves in the free space like two-component spinor field ( Berry, 2004).

Compare the intensity-polarization patterns of the beams before and after in their isomomentum planes. The pattern of the beam before is a disc with radius for a pinhole diffracted or Gaussian beam. The pattern of the beam after changes continuously its intensity distribution with the resolution . It develops as disc with radius , after that forms one ring — the Lloyd pattern, which further splits by one dark ring with radius — the Poggendorff pattern.

We picture these patterns in the plane of the CRE coordinate system. The beam before is symbolized by a green circle and the polarization axes of the ring points are indicated by blue arrows.

The case when the beam before is scalar beam is depicted below for both types of ring patterns:

In the case when the beam before is linearly polarized, the ring patterns preserve the polarization distribution, while the intensity changes along the ring. The BHS-transformation predicts that the intensity varies with . Both variables are the azimuthal angle coordinates in the standard spherical coordinate system presentation of ; belongs to the points from the ring and belongs to the polarization axis of the beam before. The Lloyd patterns when takes values of and are presented bellow.

If the beam before is circularly polarized, the BHS- transformation predicts very similar, but not identical patterns, compared with these of scalar beam.