## 2 Design Principle for a Multi-Focal MFZP  
Conventional MFZP implementation is carried out by selectively opening or closing the individual paths of light so that they may result in constructive or destructive interference at the specified focal point, respectively. Thus, assuming that the incident light is monochromatic and has a uniform phase front at the MFZP plane, i.e., \(z=0\), the radii of the individual annular slits can be determined by the following formula [27]:  
\[r_{n}=\sqrt{(n+\alpha)\lambda_{0}f_{0}+\frac{(n+\alpha)^{2}\lambda_{0}^{2}}{4} }\ (n=1,2,\ldots,2N), \tag{1}\]  
where \(r_{n}\) denotes the inner radius of an annular slit opening for an odd integer \(n\) or its outer radius for an even integer \(n\), \(N\) is the total number of annular slits, \(\alpha\) is the initial phase-offset constant, \(f_{0}\) is the focal length, and \(\lambda_{0}\) is the wavelength of incident light. Thus, this MFZP produces a hot spot at \(z=f_{0}\) via all constructive interference. This design principle can be reformulated in a reciprocal manner, because without loss of generality, the propagation of electromagnetic waves is \"reciprocal\" in a simple medium [28]. In other words, if one puts a virtual point source (VPS) at \(z=f_{0}\), where all the light is initially focused down, it radiates a spherical wave towards the MFZP plane. This phase-conjugated or backward-directed spherical wave eventually produces a phase distribution at \(z=0\) in relation to the initial phase offset value \(\alpha\). Specifically, if this phase distribution is quantized into 0 and \(\pi\) in a binary manner based on the so-called phase selection rule (PSR), subsequently making the 0 and \(\pi\) regions open and closed, the resultant pattern of the annular slits is the same as specified by Equation (1) for a mono-focal MFZP. Building upon this fundamental principle of the PSR via a VPS as well as the principle of superposition, one can readily design a multi-focal MFZP if one puts multiple VPSs at different focal points to obtain the resultant phase distribution at \(z=0\). In principle, the rest of the procedure is identical to the mono-focal MFZP case discussed above. We visualize the method in Figure 1.  
Moreover, it is worth noting that, depending on the depths of the multiple focal points, the intensities of the individual spherical waves at the MFZP plane, i.e., their influences on the resultant phase distribution, decrease based on the inverse-square law. Thus, when designing a multi-focal MFZP, one should also adjust the intensity levels of individual VPSs according to the intensity levels at the desired focal points. Consequently, considering that the quantized amplitude pattern can considerably vary in a binary manner based on the PSR, given the choice of the complex amplitudes of the VPSs, i.e., their intensity levels and initial phase-offset values, it is necessary to optimize them from the perspective of both multi-focal functionality and fabrication feasibility. In particular, we chose to use the particle swarm optimization (PSO) method [29] for this optimization procedure, although it is not necessarily the only method available. In addition, we carried out all the numerical calculations and optimizations using the commercial mathematical software Matlab (R2016b, MathWorks, Natick, MA, USA).  
Figure 1: Design principle for a multi-focal MFZP: (i) Generation of a phase profile from the VPSs placed at different focal points. (ii) Application of the PSR, selectively taking the sections where constructive interferences take place. (iii) Creation of an on–off (open–closed) metallic slit pattern based on the binary quantization of the phase profile.