A subwavelength spot and a three-dimensional optical trap formed by a single planar element with azimuthal light

The generation of subwavelength spots smaller than the Abbe diffraction limit has attracted great interest due to the various applications in many fields, such as high-density optical data storage and particle manipulation. Planar optics that can miniaturize conventional refractive optics have become increasingly attractive. In this work, we first formed a subwavelength bright spot and a three-dimensional optical trap under the illumination of an azimuthally polarized (AP) beam by only a single planar element, a spiral zone plate (SZP). Initially, the SZP was proposed as a computer-generated hologram to generate optical phase singularities. However, the SZP in this work was used to focus and modulate the incident AP beam with a vortex phase simultaneously. Therefore, no additional vortex phase modulating element was introduced in our method. The SZP has an ultra-long focal length of 250λ for a numerical aperture (NA) of 0.95 and an incident wavelength of 632.8 nm. The generated spot is purely transversely polarized with a lateral full width at half maximum (FWHM) of 0.43λ beyond the diffraction limit of 0.54λ. The generated focal field formed a stable optical trap for a Rayleigh dielectric particle in three dimensions.


Supplementary
The fabrication errors will affect the performance of the SZP, especially the radius fabrication deviation. In this supplementary, the influence of the radius fabrication deviation on the spot size is investigated.
For a circularly symmetrical device such as a FZP, the radius fabrication deviation will change the widths of the concentric belts. The influence of the radius deviation can be investigated by randomly changing the belt widths. However, the SZP is planar spiral and no longer circularly symmetrical. Consequently, the influence of the radius deviation of the SZP cannot be directly researched by this method.
We introduce the radius deviation into the calculation for the SZP by two different modes.
The transmittance function of the pth-order phase SZP is expressed as equation (2) in the main body of the article. The first deviation introduction mode is implemented by adding a random radius deviation to the radial coordinate r in equation (2) of every calculated sample point of the SZP. The added random radius deviation of every sample point obeys the uniform distribution with ±∆r max , where ∆r max is the maximum radius deviation. The dependence of the spot size on the maximum radius deviation ∆r max by this mode is shown by the line with triangular markers in Fig. S1(b). All the other calculation parameters are the same as those in the main body of the article. The markers on the line are the calculated points and the value of each point is the average of 10 calculations. Clearly, the FWHM of the generated spot increases as the increasing of ∆r max . When ∆r max = 50 nm, the FWHM of the generated spot is 0.437λ which is enlarged by 1.7% than the spot of 0.43λ. When ∆r max of mode 1 is larger than 50 nm, the spot size enlarges rapidly. The second deviation mode is similar to the method for a circularly symmetrical device.
The SZP of N max is divided into 2N max +2 zones, as shown in Fig. S1(a). The radius of each zone equals the radius of each concentric belt of the FZP which has the same NA and N max as the SZP. The radial coordinate r in each zone is added a random radius deviation ∆r i , where i ranges from 1 to 2N max +2. Fig. S1(a) shows an example of dividing a SZP of N max = 1 into 2N max +2 = 4 zones. The added random radius deviation ∆r i for each zone obeys the uniform distribution with ±∆r max , where ∆r max is the maximum radius deviation. The dependence of the spot size on the maximum radius deviation ∆r max by mode 2 is shown by the line with square markers in Fig. S1(b). The markers on the line are the calculated points and the value of each point is the average of 10 calculations. The variation of the spot size along with the radius deviation of mode 2 is smaller than mode 1. When ∆r max = 50 nm, the lateral FWHM of the generated spot is 0.436λ which is enlarged by 1.4% than the spot of 0.43λ. Similar to the results of mode 1, the spot size enlarges rapidly when ∆r max is larger than 50 nm. It is reasonable to choose 50 nm as the largest tolerated fabrication deviation of the radius.
For a commercial electron beam lithography system, this tolerance 50 nm is relatively easy to achieve. For example, the minimum feature size that the EBPG5200 of Raith GmbH can achieve is less than 8nm. Therefore, the generated sub-diffraction spot is insensitive to the fabrication errors of the SZP.