Evolution of a conically diffracted Gaussian beam in free space

Summary form only given. A beam of light propagating along the optic axis of a biaxial crystal will be transformed to a cone of light and emerge from the crystal as a hollow cylinder as predicted by W. R. Hamilton in 1832 [1] and experimentally observed shortly thereafter by H. Lloyd [2]. Recent renewed interest in the phenomenon has led to the development of a more complete theory, many aspects of which have yet to be experimentally observed.In this contribution, we compare our observations on the free space evolution of a conically diffracted Gaussian beam with the recently advised paraxial theory [3]. The theory was successfully tested for four KGd(WO4)2 biaxial crystals of lengths: 7.40 mm, 16.94 mm, 19.40 mm, and 24.50 mm, in a single-crystal conical diffraction configuration. A collimated beam from a diode laser with λ=635nm was focused to a spot of Ȧ0=13.6 μm (1/e value). For each conical diffraction crystal (CDC) the longitudinal shift, ǻ, the ring radius, R, and the position of the spot with the highest axial intensity, Zf, were measured using a Spiricon SP620U beam profiler on a mechanical travel translation stage. The longitudinal shift was given by ǻ=L(1-1/n2) where n2 is one of the refractive indices of the CDC and had values of 3.78 mm, 8.66 mm, 9.92 mm and 12.52 mm. The measured values were found to be 3.4 mm, 8.6 mm, 9.7 mm and 12.75 mm respectively. The ring radii are proportional to the length of the crystal, L, and are found using R=AL, where A is the semiangle of the cone of conical diffraction. The theoretical values were found to be: 130.23 μm, 298.14 μm 341.33 μm and 431.16 μm for each crystal. The measured values were 130μm, 293μm, 335μm and 430μm respectively. Where λ is the wavelength of the incident beam and Ȧ0 is the radius of the incident beam at the beam waist. The theoretical Zf values were found - o be 14.29 mm, 32.72 mm, 37.46 mm and 47.32 mm. The measured values were 16.35 mm, 33.80 mm, 33.45mm and 45.25mm respectively.In addition the evolution of the beam in free space was found by taking images at intervals of 0.5 mm over a range of -66 mm using CDC3 of length 19.4 mm and is shown in Figure 1. All of the reported values were found to be in good agreement with the paraxial theory derived by M. V. Berry [3] and the cross section was found to be in good agreement with the theoretically predicted image produced in [4].