Dual mirror evolutions of Airy beams propagating through uniaxial crystals induced by added spherical phase

Airy beams have received considerable attention due to their unique features on propagation, including non-spreading, self-healing, self-accelerating, and parabolic trajectories. Here in this work we study the propagation of linearly polarized Airy beams with an added spherical phase in uniaxial crystal orthogonal to the optical axis. Based on the beam transmission theory in uniaxial crystals, the analytical expressions for the intensity distribution of the beams in different view planes are derived. Numerical calculations are performed and some novel propagation features are presented graphically. It is shown that the Airy beam with an added spherical phase remains linearly polarized but cannot keep other properties unchanged during propagation in uniaxial crystal. Such a beam maintains its intensity profile in the near-field, then with the propagation distance increasing, converts into the Gaussian-Airy beams with different orientations at two specified distances which are codetermined by the extraordinary and ordinary refractive index of the crystal (namely ne and no) and the radius of the spherical phase, and most impressively, forms a mirror-like reflection profile in the far field, i.e., the intensity pattern in the far field returns to the initial Airy beam profile while its orientation on the transversal plane is reversed along the bisector line of the second and fourth quadrant. Note that the intensity pattern successively experiences two mirror transformations along the x and y coordinate axis when passing through these two critical positions, which can give rise to the mirror reflection effect for the whole Airy beam. Moreover, we further demonstrate that the sequences of these two mirror transformations are in close relation with the relative size between ne and no. Therefore, the results obtained in this paper reveal new propagation features in anisotropic medium of Airy beams with added spherical phase and provide a novel route to controlling propagation properties like the pattern profile and orientation of the Airy beams through choosing appropriate anisotropic materials and the radius of the spherical phase factor. Considering that it is easy to obtain an Airy beam with an added spherical phase which can be realized with an Airy beam through an ideal lens, our investigation may lead to potential applications in many fields where the ability to change profile and orientation of the intensity pattern and the ability to determine the refractive index of anisotropic medium are both required.