Internal conical refraction of light beams in biaxial gyrotropic crystals

Conical refraction: fundamentals and applications

“…These results describe in detail all the phenomenology associated with CR, which were demonstrated by Fève et al with spheres of KTP [28]. The Beslky and Khapalyuk theory of CR was explored in detail by Belsky and Stepanov [29] (including gyroscopic crystals [30]), who recognized the relevance on the CR beam evolution of the ratio between the ring radius and the waist radius of the input beam, and also by Belafhal for the asymptotic case of R 0 ≫ w 0 [31], where R 0 is the geometric approximation of the CR ring radius and w 0 is the waist radius of the input beam.…”

“…Besides, we have shown that the radius of the PDR is smaller than the optical geometric approximation of the CR ring radius R 0 , by approximately half the waist radius of the input beam (−0.541w 0 in Table 7.1). All previous related works [27,29,31,32,74] were performed considering that the radius of the PDR exactly coincided with R 0 . The reported results show that the PDR is enclosed by higher intensity walls both in the radial as in the axial directions, i.e., it is a toroidal dark-focus in all three dimensions, at variance with other light beams possessing only radial confinement, such as Laguerre-Gaussian modes.…”

“…That is the reason why the diffractive theory of CR, capable of predicting all the observed phenomena, had to wait until the last quarter of the last century. This theory was presented by Belsky and Khapalyuk [26,27] and then reformulated in an elegant way by Berry [32]. Recently, we have shown that the reformulation of the diffractive theory of CR carried out by Berry can be used to predict the beam evolution of light propagation in biaxial crystals along any direction and not only along the optic axis [52] and also in case of non-homogeneously polarized beams [53].…”

“…In 1978 Belsky and Khapalyuk presented the holy grail of CR: the full diffractive theory of both external [26] and internal CR [27] in biaxial crystals. This work is very related to Lalor's calculations but the equations at which Belsky and Khapalyuk arrived were much more compact.…”

“…ρ 0 is the ratio between the geometric ring radius of the CR ring R 0 and the waist radius of the focused input beam w 0 , i.e., ρ 0 ≡ R 0 /w 0 . We would like to note that this transformation is assumed in the formulation of Belsky and Khapalyuk [26,27]. It is straightforward to demonstrate that for a given vector v = vn with |n| = 1, the evaluation of e iv( n· σ) givesÎ cos(v) + i(n · σsin(v)), where σ = (σ 1 ,σ 2 ,σ 3 ) is the vector of Pauli matrices.…”