To solve the problem that the traditional method can only produce spherical focused spots along the optical axis, we propose a method to generate spherical focused spots in any arbitrary spatial direction in a 4Pi focusing system, which consists of two opposing high numerical aperture objective lenses with the same focus. Spherical focused spots with equivalent three-dimensional spatial resolution have important applications in optical microscopy and metal particle capture. In particular, these spots can trap metal particles at resonant wavelengths, which is because the enhanced axial gradient force and the symmetry of the 4Pi focusing system can offset the axial scattering and absorption forces, making it possible to stabilize the trapping of resonant metal particles and precisely control the motion trajectory of metal particles. Spherical focused spots should be generated at any spatial position to capture resonant metal particles at arbitrary spatial positions. To our knowledge, this is the first time that controllable spherical focused spots can be obtained at an arbitrary spatial position. The proposed method features greater flexibility than traditional approaches, making it highly valuable for applications involving nanoparticle capture at arbitrary spatial locations.

We present a method to generate spherical focused spots with the specified spatial direction and spacing in a 4Pi focusing system using dipole antenna radiation fields generated by de-focusing. The method involves placing the spatial dipole antenna with predefined lengths and polarization direction at the focal point of the 4Pi focusing system and solving the inverse problem to determine the input field on the objective pupil plane that generates spherical focused spots. By utilizing the field on the pupil plane and selecting the appropriate length of the dipole antenna, spatial spherical focused spots can be obtained.

Firstly, the number of generated spatial spherical focused spots is related to odd or even multiple of half wavelength (Fig. 2). When the length of the dipole antenna L is an odd multiple of half wavelength, two same intensity spherical spots symmetrical at the center of the focus are formed in the set spatial direction. When L is an even multiple of half wavelength, three spherical focused spots with the equal size are formed, with one high-intensity spot at the focus and two lower-intensity spots symmetrically arranged. Since the distance between spatial spherical focused spots is calculated to be equal to L, the distance of spatial spherical focused spots can be easily adjusted by changing the parameter L. Meanwhile, arbitrary spatial directions of spherical focused spots are created to demonstrate the flexibility of the proposed method (Figs. 4 and 5). It is observed that the direction of the spherical focused spots is consistent with the polarization direction of the dipole antenna. Finally, we investigate the normalized input field Eiρ,φ required to create the spatial spherical focused spots (Figs. 6 and 7). It is evident that the polarization direction of the input field is determined by the dipole antenna parameter φ0, and the dipole antenna parameter θ0 determines the spatial rotation angle of the input field.

We present a simple and flexible method for generating spherical focused spots of prescribed length and controllable spatial orientation. By focusing the electromagnetic field radiated by a virtual dipole antenna in reverse at the focal point of a 4Pi focusing system, spherical focused spots with specified characteristics can be conveniently obtained. The simulation results show that the number of spherical focused spots is related to odd or even multiple of half a wavelength, and the distance between spherical focused spots is adjustable and only depends on the antenna length L, while the spatial direction of spherical focused spots is controllable and depends on the antenna parameters θ0,φ0. Furthermore, all spherical focused spots generated by the optical antennas are of the same size, with a full width at half maximum (FWHM) of 0.459λ. The generated spatial spherical focused spots have potential applications in precise multi-point trapping of spatial nanoparticles with full degrees of freedom, showing broad prospect in optical micro-manipulation.