Objective The traditional focusing device is restricted by the Abbe diffraction limit. This means that the spatial resolution cannot exceed its theoretical minimum value of 0.5 lambda/ NA, where lambda is the working wavelength and NA is the numerical aperture. Existing methods to break the diffraction limit require a near-field environment, which is insufficient for far- field super- resolution imaging in the optical sense. The principle of optical super-oscillation states that it is theoretically possible to produce a super-resolution spot of arbitrary smallness by rationally modulating the wavefront of incident light. Optical super- oscillation has been extensively studied by researchers in super- resolution optical lenses, and this principle enables the experimental realization of far-field super-resolution focusing. However, the optical field regulation of the superoscillation lens depends on precise nano-processing technology. Additionally, the fabrication cost and complexity limit the device to a small size. Thus, we propose a method to generate the far-field super-resolution optical field based on the spatial light modulator. The design of the far- field super-resolution focusing device is based on the super-oscillation principle, with the binary particle swarm optimization algorithm and the angular spectrum diffraction theory combined. The generated focal spot full width at half maximum ( FWHM) is smaller than the diffraction limit, which can be employed to construct the far-field super-resolution optical field. Methods The device is designed based on the super- oscillation principle and adopts eight-value phase control for circularly polarized light with a wavelength of 632.8 nm. The two-dimensional phase distribution of the device is optimized using the binary particle swarm optimization algorithm and angular spectrum diffraction theory. This optimization helps obtain the optimal phase of the mask and its corresponding characteristic parameters. The device is composed of a series of concentric ring belts, each with 8 mu m width, which is equal to the size of spatial light modulator (SLM) pixels adopted in subsequent experiments. To obtain an optimized phase mask, we calculate the phase of each ring belt and generate a grayscale image based on the SLM phase control characteristics. Additionally, to verify the focusing performance of the designed device, we design and build a construction and measurement system for the far-field super-resolution optical field. We measure the characteristic parameters of the super-resolution optical field using an objective lens combined with a complementary metal oxide semiconductor (CMOS) camera. The motorized linear translation stage is moved to obtain the two- dimensional optical field distribution at different positions. Finally, an image processing algorithm is then utilized to extract the key focusing parameters of the focal spot, leading to a three- dimensional intensity distribution of the optical field. Results and Discussions First, the corresponding grayscale images are generated based on the phase of each ring belt of the super-oscillatory mask obtained from the optimized design ( Fig. 3). Next, the design results of the optical field are calculated by adopting the angular spectrum diffraction theory (Fig. 4). An experimental platform is then set up, and the super- oscillatory mask is loaded onto the liquid crystal screen of the spatial light modulator. Finally, the optical field is scanned and tested within the range of Z= 185.00 mm to Z=195.00 mm. The scanning step Delta Z is 0.05 mm, and the intensity distribution of the optical field is obtained. Experiment and theoretical results demonstrate excellent agreement, and the transverse FWHM at the focal length of the focal spot is 22.384 mu m, which is below the diffraction limit (0.5 lambda/ NA, 23.732 mu m), with far-field super-resolution focusing achieved (Fig. 6). Along the propagation direction, the vertical FWHM is 6.029 mm, creating an optical needle (Fig. 7). The device is easy to operate and does not require complex processing. Conclusions To solve the problem of traditional focusing devices are constrained by the diffraction limit, we propose a method for constructing a far-field super- resolution optical field with eight-value phase control based on the optical super-oscillation principle. By adopting particle swarm optimization and angular spectrum diffraction theory, we design a far- field super- resolution focusing device for circularly polarized light with a wavelength of 632.8 nm. This is achieved by loading a super- oscillation phase mask onto the liquid crystal screen of a spatial optical modulator. By adjusting the phase of the incident optical field, the device generates an optical needle with the vertical FWHM of 6.029 mm. The FWHM at the focal length of the focal spot is lower than the diffraction limit, thus achieving far-field super-resolution focusing. This method can be applied to the visible bands and extended to other optical bands, providing core focusing devices for optical microscopy, optical imaging, and other optical applications.
