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Timestamp: 2019-04-25 02:12:00+00:00

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We propose a paraxial dual-cone model of conical refraction involving the interference of two cones of light behind the exit face of the crystal. The supporting experiment is based on beam selecting elements breaking down the conically refracted beam into two separate hollow cones which are symmetrical with one another. The shape of these cones of light is a product of a ‘competition’ between the divergence caused by the conical refraction and the convergence due to the focusing by the lens. The developed mathematical description of the conical refraction demonstrates an excellent agreement with experiment.
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Fig. 1 Diagram of the optical setup used to observe conical refraction. A laser source is focused by a lens (Lens 1) and then passed through a CRC. A second, translatable, lens (Lens 2) can then be used to image the beam onto a screen or CCD. Lens 2 is not always needed since it can be possible to place a CCD close enough to the crystal facet. A and B represent positions in the optical train where we can place various optical components such as a pinhole or opaque spot. The images on the right are intensity profiles of the beam at various points after emerging from the crystal. The Lloyd ring exists in the focal image plane (FIP) of the beam. Before and beyond this point the beam exhibits symmetry changing from a double ring with a dark ring between them (Poggendorff’s rings) to an intense spot with a Bessel-like nature (Raman spot).
Fig. 2 Theoretical and experimental axial distributions of the CR beam intensity. (a) Axial distribution of the CR beam intensity computed numerically from the dual-cone model. (b,c) The individual cone components of CR beam |C1|2 and |C2|2. (d) Stack of CCD images of unimpeded CR beam. (e,f)Same experiment but using a pinhole (e) and an opaque spot (f) immediately after the CRC to split the beam into two constituent cones. Images d,e&f were taken over 100mm in 1mm steps with an exposure time of 0.2 ms.
Fig. 3 Scheme of conical refraction.(a) Internal conical refraction . (b) External conical refraction . (c) Internal conical refraction of slightly converging beam. (d) Conical refraction of strongly convergent beam. The black dotted lines in (c,d) indicate Hamilton’s ray trace. (e) Experimental intensity distribution of the beam immediately before the entrance face of the CR crystal. (f) Experimental intensity distribution of the beam immediately after the exit face of the CR crystal. (g) Experimental intensity distribution of the beam in the Lloyd plane. The white dotted lines in (e-g) guide the eye and indicate the diameter of the dark ring in the Lloyd plane and it’s double (i.e. the diameter of the cone C2 on the exit face of the crystal).
Fig. 4 Proof of the interference nature of the Lloyd ring. Left, diagram of the two cones C1 and C2 (blue and red) showing the principle of changing the optical path difference for the interfering cones with introduction of a glass slide. This glass slide could be tilted on an axis perpendicular to the axis of propagation to introduce a varying phase delay. Right, experimental and theoretical distribution of the CR beam intensity in the Lloyd plane. (a) CCD images from the Lloyd plane showing how the intensity distribution is changed as the phase difference increases. (b) Numerical computation of the intensity distribution in the Lloyd plane with the dual-cone model with an extra phase shift Δφ introduced for a section of the outer light cone C2. The radial intensity profile cross-sections are shown in white. (i) Without extra phase shift the distribution is unaltered. (ii) With extra phase shift Δφ = π/2 only one ring can be observed in the Lloyd’s plane. (iii) With Δφ = π, intensity profile of the Lloyd’s double-ring is inversed. (iv) With Δφ = 3π/2the intensity of both rings in the Lloyd’s plane is equalized. (v) With Δφ = 2π, numerically computed intensity profile of the Lloyd’s double-ring is obviously the same as without extra phase shift. Experimentally, the effects of diffraction on the edge of the glass slide are visible. In the numerical picture, schematic glass slide marks the region where the phase shift Δφ was introduced.

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