## 3 A Monochromatic, Multi-Focal MFZP  
To demonstrate a proof of principle for multi-focal functionality, we first designed an example MFZP capable of producing multiple hot spots at three different points with identical peak intensities for monochromatic incident light. We chose the design parameters as the following: we set the diameter of the target MFZP to 50 \(\upmu\)m and the three foci to \(f_{1,2,3}=10,\,15,\,\) and 20 \(\upmu\)m. We assumed that the foci were on the central axis of the MFZP, i.e., the \(z\)-axis, and that the wavelength of the incident light was 650 nm. For simplicity, we initially chose the indicators of the PSO algorithm from two perspectives: (1) how accurately the three foci were formed at the desired locations in terms of \(\Delta f_{m}\); and (2) how evenly the peak intensities of the three hot spots were determined in terms of \(\Delta l_{m}\). However, it is worth noting that the optimization procedure can further be extended in various additional aspects if necessary. We set the penalty parameters based on those two simple criteria and carried out the optimization procedure based on the PSO algorithm until both criteria were met within the tolerance values, such that the relative tolerance values for \(\Delta f_{m}\) and \(\Delta l_{m}\) were \(\pm 0.5\%\) and \(\pm 3\%\), respectively. In Table 1, we summarize the target design parameters as well as the resultant parameters obtained after going through the application of the PSR via the VPSs along with the PSO algorithm. In particular, it is worth noting that we set the minimum step for the slit-width variation, i.e., the slit resolution, to 10 nm, in consideration of the fabrication resolution of the in-house focused ion beam (FIB) system. In Figure 2, we illustrate the resultant pattern of the designed MFZP and the corresponding numerical result regarding its multi-focal functionality. It is worth noting that the intensity level at \(f_{3}\) was regarded as the reference value. From the result, one can clearly see that it could produce three hot spots at the expected locations with sufficiently identical peak intensity levels. The individual locations of the three focal points were 9.95, 15.04, and 19.99 \(\upmu\)m, all of which were within the specified tolerance. Considering that we simplified the optimization criteria for this MFZP for the purpose of a proof of principle demonstration, there remains room for improvement in aspects other than the location accuracy and intensity uniformity. For example, in the above simulation results, the side lobe suppression ratio was estimated to be \(-6.12\) dB. One can improve this ratio significantly if one extends the optimization procedure with an additional criterion. Notwithstanding this, we emphasize that the proposed method, based on the application of the PSR via the VPSs along with the PSO algorithm, was very effective and powerful in systematically designing an MFZP with multi-focal functionality for monochromatic incident light. Based on the designed pattern shown in Figure 2a, we fabricated a real MFZP and characterized it experimentally. The fabrication procedure was as follows: We first uniformly deposited an Au metal layer of 100 nm thickness on a glass substrate, using an electron beam evaporation method [30]. We implemented the designed pattern via the FIB milling process [18; 31]. In Figure 3, we illustrate the schematic diagram of the experimental arrangement for the characterization of the fabricated MFZP. It comprised a light source, an MFZP sample, and an imaging device. We used a diode laser (VFL-250, Fibretool, Shanghai, China) operating at 650 nm as the light source to irradiate the MFZP sample. It is worth noting that the diode laser could be replaced by