The Talbot effect has been studied intensively in optics, acoustics, electron microscopy, X-ray, and Bose-Einstein condensates. There are numerous important applications of the Talbot self-imaging phenomenon in modern optics, such as in optical measurement, array illumination, lithography, color digital holography, and X-ray imaging technology. Uniaxial crystal is a typical kind of anisotropic media which has been widely used in different optical devices, so the propagation of light beams in uniaxial crystals is also an important topic in the field of optics and optoelectronics. However, no research has studied the behavior of periodic light fields in uniaxial crystals, especially the Talbot effect in uniaxial crystals. Therefore, in this work, we theoretically investigate the Talbot effect in uniaxial crystals orthogonal to the optical axis to prove that the Talbot self-imaging phenomenon can also be observed in anisotropic media. Our results can extend the studies of the Talbot effect to the field of anisotropic media and may improve our understanding of the transmission properties of periodic light fields.

Based on the beam transmission theory in uniaxial crystals, we successfully derive theoretical formulas to investigate the Talbot effect due to the propagation of the two-dimensional (2D) orthogonal periodic light field along the direction perpendicular to the optical axis of the uniaxial crystal (z-axis) when the optical axis of the uniaxial crystal is along the x-axis. When a 2D periodic object is illuminated by an x-polarized coherent uniform beam, the propagation is described by anisotropic diffraction. With the help of the Fourier transform and angular spectrum representation, we can derive the self-imaging conditions of the Talbot effect and the expression of the conventional Talbot distances. We have also performed numerical simulations to observe the anisotropic propagation of two special 2D orthogonal periodic optical fields (2D sinusoidal grating and 2D checker grating) in the rutile crystal. In virtue of the structures of these periodic optical fields, the Talbot self-imaging phenomenon can be observed in shorter propagation distances. The Talbot images obtained by these two kinds of light fields in the rutile crystal are presented and analyzed.

We find the self-imaging conditions of the Talbot effect. The expression of the conventional Talbot distances depends on the ratio of the ordinary refractive index to the extraordinary refractive index in the uniaxial crystal (n_o/n_e) and the ratio of the period of the x direction to the period of the y direction in the 2D periodic object (p_x/p_y), as expressed by Eqs. (13) and (14). We have performed numerical simulations to observe the Talbot effect of the 2D sinusoidal grating and the 2D checker grating due to the anisotropic diffraction in the uniaxial crystal. In both cases, the Talbot images which are the same as their original gratings can be generated at reduced Talbot distances [Fig. 2 (a) and Fig. 5 (a)], and the complementary images of their original gratings can be generated at the reduced half-Talbot distance [Fig. 2(b) and Fig. 5(b)]. The Talbot sub-images with a half shift in space, whose intensity periods are halved from that of their original gratings and phase periods are the same as the period of the original gratings, are generated at the reduced quarter-Talbot distance and the reduced three-quarter-Talbot distance, as shown in Figs. 2(c)-(f) and Figs. 5(c)-(f). For the case of the 2D sinusoidal grating propagating along the z-axis (perpendicular to the optical axis of the uniaxial crystal), we have plotted the contrast variation of the intensity pattern and the phase contrast in a reduced Talbot distance (Fig. 3). The light intensity reaches its maximum value at the reduced Talbot distance and the reduced half-Talbot distance. At the reduced quarter-Talbot distance and the reduced three-quarter-Talbot distance, the light intensity reaches its minimum value. For the phase contrast, the maximum values are found at the reduced quarter-Talbot distance and the reduced three-quarter-Talbot distance.

Based on the paraxial theory of light propagation in uniaxial crystals, we suppose that the optical axis is along the x-axis, and the propagation is along the z-axis. Furthermore, we have theoretically investigated the Talbot self-imaging phenomenon. When a 2D orthogonal periodic object is illuminated by the x-polarized coherent uniform beam, the anisotropic diffraction leads to the self-imaging conditions of the Talbot effect depending on n_o/n_e and p_x/p_y. If the 2D orthogonal periodic optical object meets certain conditions, the Talbot images can be generated in shorter propagation distances, which means that the Talbot effect can be observed at reduced Talbot distances. Finally, for an incident light field linearly polarized at an arbitrary direction, the Talbot distance of the x-polarized component caused by the anisotropic diffraction does not equal that of the y-polarized component caused by the isotropic diffraction. Therefore, the repetition of the periodic light field can only occur at positions corresponding to the common multiples of these two distances. In other words, the Talbot distance corresponding to the arbitrarily linearly polarized case should be increased.