## 4 A Multi-Chromatic Mono-Focal MFZP
In addition, the relative peak intensity levels of blue and green colors to that of red were 0.73 and 0.76, respectively, which was due to the fact that we did not put any indicator in the PSO procedure that took into account the relative peak intensity levels among colors. In order to adjust  
Figure 5: Numerical simulation of the monochromatic multi-focal MFZP specified in Table 1, regarding the normalized intensity distribution along the \(z\)-axis with respect to incident wavelength. them further, one can take the recourse of optimization by putting additional tolerance on \(\Delta I_{m}\), as was done in the preceding section. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & & & & Tolerance \\ \hline Parameters & Focus Length & Wavelengths & Diameter & Slit Resolution & \multirow{2}{*}{0.1\%} \\ \cline{1-1} \cline{5-5}  & 15 \(\mu\)m & 473, 532, 660 nm & 50 \(\mu\)m & 10 nm & 0.1\% \\ \hline \multicolumn{5}{c}{Optimized Parameters} \\ \hline Parameters & Relative Intensities (\(I_{1}\), \(I_{2}\), \(I_{3}\)) & Initial Phase-Offset (\(\phi_{1}\), \(\phi_{2}\), \(\phi_{3}\)) & Position \(|\Delta f_{m}|\) \\ \hline Value & 1, 1, 1 (Fixed) & 2.00 \(\pi\), 0.92 \(\pi\), 1.38 \(\pi\) & \textless{}0.07\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Simulation parameters for a multi-chromatic, mono-focal MFZP. Figure 6: Simulation results for the multi-chromatic mono-focal MFZP specified in Table 2: (**a**) Shape of the calculated MFZP pattern. (**b**) Calculated intensity profiles on the center axis of the MFZP, i.e., the \(z\)-axis, for RGB beams. (**c**) Calculated intensity pattern formed through the MFZP for RGB beams. Based on the designed pattern shown in Figure 6a, we fabricated an MFZP with multi-chromatic mono-focal functionality, for which we used the electron beam evaporation and the FIB milling processes, as discussed in the preceding section. We characterized it experimentally, using the same setup as illustrated in Figure 3 in which we replaced the 650 nm laser diode with an RGB laser system. The RGB laser system was constructed by combining individual red (Flamenco 300, Cobolt, Solna, Sweden), green (Verdi G2 SLM, Coherent, Palo Alto, CA, USA), and blue (Excelsior 473, Spectra-Physics, Santa Clara, CA, USA) laser beams operating at 660, 532, and 473 nm, respectively, using dichroic mirrors. In Figure 7a,b, we illustrate the SEM image of the fabricated MFZP and the images of the hot spots obtained using the optical microscope for individual color beams as well as for the combined RGB beams at the same location, i.e., \(z=\) -15 \(\upmu\)m, respectively. One can see that the fabricated MFZP effectively produced hot spots at the same location for all the RGB beams. In particular, when the individual RGB beams were illuminated on the MFZP at the same time, one can clearly see a single bright hot spot in white color at the center, which means that the three hot spots in RGB colors were formed at the same location. Owing to the limited sensitivity of the CMOS sensor array of the optical microscope system we used, we could not resolve out even finer intensity distributions than detected, except for some low intensity side lobes in the transverse plane. Notwithstanding this, we emphasize that overall the experimental results were in good agreement with the numerical results presented in Figure 6.