## 4 A Multi-Chromatic Mono-Focal MFZP  
It is worth noting that a conventional MFZP has a high chromatic aberration such that the effective focal length varies with a wavelength with a negative slope [32]. In other words, the longer  
Figure 4: Experimental characterization of the fabricated monochromatic multi-focal MFZP: (**a**) SEM image of the fabricated MFZP. (**b**) Images of the transmitted light through the MFZP with respect to the longitudinal position taken by the optical microscope system for monochromatic incidence at 650 nm: (i) \(z=0\)\(\upmu\)m, (ii) \(z=10\)\(\upmu\)m, (iii) \(z=15\)\(\upmu\)m, and (iv) \(z=20\)\(\upmu\)m. (**c**) Comparison between the experimental and numerical results regarding the intensity profile on the center axis of the MFZP, i.e., the \(z\)-axis. (**d**) Whole spatial distribution of the transmitted light in both longitudinal and transverse directions. the wavelength, the shorter the focal length. This feature can readily be seen from the relation between \(f_{0}\) and \(\lambda_{0}\) in Equation (1) and can also be verified in the simulation results shown in Figure 5, where we numerically analyzed the focusing characteristics of the monochromatic multi-focal MFZP designed for 650 nm in the preceding section, extending the operating wavelength range from 600 to 700 nm. Although there were a number of side lobes of low intensity levels, we noted that there were three main traces with negative slopes that could lead to significant hot spots at three different locations for a fixed wavelength. Considering the negative slopes, one can also find three different wavelengths of incident light that can simultaneously lead to producing a hot spot at a fixed distance. Specifically, if one draws a horizontal line in Figure 5, it will intersect the three main traces at three different wavelengths. This means that a monochromatic multi-focal MFZP can in principle also function as a multi-chromatic mono-focal MFZP that simultaneously produces a hot spot at an identical location for multiple wavelengths, e.g., multi-color or RGB beams. In order to produce such a hot spot at a desired location for desired multiple wavelengths, we once again could exploit the PSR via the VPSs discussed in the preceding section. In this case, all we had to do was place all the VPSs at an identical location but with different wavelengths. To demonstrate a proof of principle for multi-chromatic, mono-focal functionality, we designed and fabricated another MFZP capable of simultaneously producing a hot spot at an identical location for RGB beams. We assumed that the wavelengths of the RGB colors were 660, 532, and 473 nm, respectively, and that the target focal point was located at 15 \(\upmu\)m away from the MFZP plane. In a similar manner to the design procedure taken in the preceding section, we carried out the application of the PSR via the VPSs along with the PSO algorithm. We note that in this case we took a single indicator for optimization only from the perspective of how similarly the three focal points were formed at the desired location in terms of \(\Delta f_{m}\) with a relative tolerance value of \(\pm\)0.1%. It is worth noting that we did not consider a second criterion on \(\Delta I_{m}\) at this stage, mainly because the incident multi-color beams do not necessarily have the same power levels. We summarize all the simulation parameters in Table 2.  
In Figure 6, we show simulation results in which one can see that all the RGB colors were focused on a nearly identical location at 15 \(\upmu\)m. The minimum side lobe suppression ratio for all the cases was -3.47 dB, which was relatively higher than that of the monochromatic multi-focal MFZP. However, we note that one can improve it significantly if one extends the optimization procedure with an additional criterion.