Surface plasmon polaritons (SPPs) are an electromagnetic mode in which an external light field interacts with the free electrons on a metal surface. When light is incident on the surface of the metal and dielectric, the collective free electrons oscillate on the surface of the metal owing to the incident light excitation effect. This causes the coupling of light and free electrons on the surface of the metal and the spread of the media surface along a conductor, which is constrained and attenuated in the vertical direction at the interface. The surface plasmon polaritons have strong near-field electromagnetic field characteristics showing strong interactions with nanostructures especially with nanomaterials composed of delicate structures. Because the diffraction limit can be exceeded and the nano-level sub-wavelength spot can be formed in the near-field region, it is of great significance to design optical near-field focusing devices. Different from the widely studied toroidal, elliptical, and helical surface plasmon focusing lenses, a surface plasmon polaritons focusing lens composed of rectangular slits is designed, and its near field mode is studied. In the case of incident light excitation, the propagating direction of the surface plasmon polaritons is always perpendicular to the long axis of the rectangular slit. When the rectangular slit rotates on the surface of the metal, it can produce an optical electrical field with a controllable geometric phase, which can be used to manipulate the near electrical field. The rectangular slit not only modulates the optical electrical field in terms of its depth, length, and width as toroidal, elliptical, and helical units do but also produces another kind of phase manipulation in terms of its axis orientation, which enriches the means of light field manipulation. In brief, a single rectangular slit generates SPPs, and then coherent interference occurs in the process of propagating towards the center of the circle. Its size and spatial position are related to the number of lens slits and the size of the lens. In this paper, finite difference time domain method is used to calculate the electric field intensity of the structure. By analyzing the finite difference time domain simulation results, we find that the lens focuses best when the size and number of the lens slits causes the interference ring to disappear. In addition, we can use the simplest coherent interference model to explain the interference bright ring pattern, with the strongest envelope at a specific location in the internal surface plasmon polaritons focusing lens. Although it is difficult to get the analytic solution of the strongest interference bright ring, the interference model used in this paper is self-consistent and can be used to verify the accuracy of the empirical formula obtained from the finite difference time domain simulation results. By adjusting the spatial rotation angle of each slit, the geometric phase of the SPPs can be generated, and the central focusing of the SPPs focusing lens can be finally realized. The focusing effect is the best when the interference ring disappears. This paper provides a new idea for the design of a surface plasmon polaritons focusing lens. The surface plasmon polaritons focusing lens in this paper connects the geometric phase to near field focusing; this gives us an enhanced understanding of geometric phase and provides a new direction for the combination of surface plasmon polaritons and topology.
