## 5 Conclusions  
We have demonstrated a monochromatic, multi-focal MFZP and a multi-chromatic, mono-focal MFZP using a novel design principle. This novel design principle is based on the PSR by implementing VPSs at the desired focal points. We utilized the PSO algorithm to determine the appropriate intensity levels and phase offset values for individual VPSs, thereby optimizing the resultant MFZP pattern from the perspective of both multi-focal functionality and fabrication feasibility. Based on the proposed design principle, we first devised a monochromatic multi-focal MFZP for 650 nm incident light, capable of producing multiple hotspots at 10, 15, and 20 \(\upmu\)m with a relative tolerance of the focal distance of <0.5% and with a relative peak intensity tolerance of <2.9%. We experimentally implemented the designed MFZP pattern onto an Au-coated glass substrate, utilizing electron beam evaporation deposition and the FIB milling processes. We characterized it with a 650 nm laser source and an imaging  
Figure 7: Experimental characterization of the fabricated multi-chromatic mono-focal MFZP: (**a**) SEM image of the fabricated MFZP. (**b**) Images of the transmitted light through the MFZP at \(z=15\)\(\upmu\)m with respect to incidence wavelength: (i) \(\lambda=473\) nm (blue), (ii) \(\lambda=532\) nm (green), (iii) \(\lambda=660\) nm (red), and (iv) combined RGB incidence (white).  
apparatus based on an optical microscope with a translation stage. We verified that the measured effective focal distances and relative peak intensity levels for individual hot spots were in good agreement with the numerically calculated results. We then extended the proposed design principle to the case of a multi-chromatic, mono-focal MFZP, which was capable of simultaneously producing a hot spot at an identical location for RGB wavelengths of 660, 532, and 473 nm. We numerically verified that it could produce a single hot spot at a focal distance of 15 \(\upmu\)m for the three RGB wavelengths with relative tolerance of the focal distance of \(<\)0.07%. We fabricated and experimentally characterized it in a similar manner. We verified that the measured multi-chromatic, mono-focal characteristics were also in good agreement with the numerical results.  
Both the monochromatic, multi-focal MFZP and the multi-chromatic, mono-focal MFZP can have various applications, such as laser micro-machining, optical trapping, biomedical sensing, confocal collimation, achromatic optics, etc. All of these invariably require novel characteristics in terms of beam focusing. Moreover, the proposed design principle based on the PSR via VPSs should be a very effective way of systematically designing various MFZP devices with novel functionality. In addition, although we only utilized the PSO algorithm for the optimization of the intensity and phase offset parameters of the VPSs, various other algorithms, such as deep learning techniques, can also be exploited in order to improve optimization performance and efficiency.  
This work was supported in part by a National Research Foundation of Korea (NRF) grant funded by the government of Korea (2017R1 DIA1B03036201); the Ministry of Trade, Industry and Energy (Project no. 10060150); and the Brain Korea 21 Plus Program.  
[PERSON] performed the experiments and numerical simulations and wrote the manuscript. [PERSON] contributed to the experiments and fabrications. [PERSON] prepared the samples and supported the experiments. [PERSON] contributed to the numerical simulations. [PERSON] supervised [PERSON]'s work. [PERSON] led and supervised [PERSON], [PERSON] and [PERSON]'s work and revised the manuscript. The authors declare no conflict of interest.