## 1 Introduction  
Fresnel zone plates (FZPs) or metallic Fresnel zone plates (MFZPs) have received considerable attention in recent years as an efficient focusing element for flat optical applications [1; 2; 3; 4; 5]. They can be used not only as a replacement for conventional focusing lenses, but also as optical elements with exquisite properties by implementing various novel patterns onto the metallic annular slit structure, such as multi-focal, multi-chromatic, and super-oscillatory lenses [6; 7; 8; 9; 10]. Among them, studies of MFZPs for multi-focal applications have recently received considerable attention. These studies have exploited various authentic mathematical formulae as generating functions for the metallic annular slit structure, such as Fibonacci, fractal, and cantor series [11; 12; 13]. While they may lead to unique multi-focal capabilities, it is hard to say whether they can rigorously produce multi-focal points at certain predetermined locations. Moreover, there have been very few attempts to investigate a systematic design method for multi-focal MFZPs even though such a method could have a huge impact on improvements in laser micro-machining, optical trapping, chemical sensing, biomedical sensing, confocal collimation, achromatic optics, etc. [14; 15; 16; 17; 18; 19; 20; 21]. Thus, we think that this field merits additional research.  
One of the simplest and most conventional ways to realize a multi-focal MFZP is to divide the area of a lens into multiple sections, such that each section is individually designed and constructed to produce a different focal point [22; 23; 24]. Whilst this spatial division can be carried out either azimuthally or radially, it can produce undesirable outcomes. The former cannot avoid polarization dependence issues, because the individual section naturally results in a different orientation to the polarization axis of incident light. In contrast, the latter incurs significant degradation in the sharpness of the focal depth [25], because the effective beam width allocated to each radial section is inevitably reduced. On the other hand, a more intricate way to realize a multi-focal MFZP is to exploit a pixelated phase pattern on the substrate via dielectric material deposition [26], such that all the combined diffractions eventually form multi-focal spots at different depths. However, such a highly pixelated phase pattern generally requires a complicated fabrication process, such as plasma-enhanced chemical deposition. This aspect substantially undermines the fundamental merits of MFZPs, which include the simplicity of both its design and the fabrication procedure.  
We here propose and demonstrate novel multi-focal MFZPs that can overcome the issues mentioned above as well as providing a simple but powerful systematic design method for them. We exploit the use of virtual point sources, placed at the desired focal points, to obtain an effective MFZP pattern of multi-focal functionality, replying to the phase conjugation principle. We can readily fine-tune and optimize the pattern, varying the complex amplitudes of the virtual point sources, i.e., their intensity levels and initial phase offset values so that the corresponding MFZP results in the best performance from the perspective of both multi-focal functionality and fabrication feasibility. We thus present all the details of the proposed systematic design method for multi-focal MFZPs and utilize the method to configure two types of MFZPs with novel functionality, including a monochromatic multi-focal MFZP and a multi-chromatic mono-focal MFZP. We fabricated them and verified their novel characteristics both numerically and experimentally.