We demonstrate our procedure through a specific design with the specifications given in Table 1. Fig. 3(a) shows the ZEMAX simulation configuration in which the surface phase with a size of 3 mm diameter represents the FZP pattern. We use the 5 terms of polynomial expansion of the surface phase as optimization parameters of the default merit function for spot radius. Besides, the EFFL operand is added to constrain the 80 mm of focal length, and the RWCE operand is used to optimize the RMS wavefront error with a target of 0; Both are considered with the weight factor of 1 shown in Table 2. The diffraction order 1 is implicit in this study. The optimized surface phase obtained with the peak to valley (PV) of 22.2 periods of 2π is performed in Fig. 3(b) and the polynomial coefficients (A_i) obtained are presented in Table 3.

Fig. 3 
Numerical simulation for the FZP pattern

We successfully apply the 5-step procedure presented in section 2.1 and evaluate the performance of the designed FZP pattern by the 3D results presented in Fig. 4. Fig. 4(a) is the ZEMAX surface phase obtained after the optimization process presented in section 2.2, then they are divided by equation (2) at a level of 2π as Fig. 4(b). The phase plate obtained in Fig. 4(c) from binarizing the kinoform surface and then obtaining the FZP pattern. Fig. 4(d) is the footprint of the FZP pattern modeled by UDA and it is used to evaluate light propagation by ZEMAX-POP as Fig. 4(e). The input Gaussian light source is implemented by a plane wave perpendicular to the light propagation direction. We show the beam shape through the FZP pattern to the focal plane and the xy plane’s light propagation. At the focal plane, we obtain the beam expressed as a delta function.

Fig. 4 
The 5-step procedure performance of the specific design FZP in section 2.2. (a) Surface phase optimization, (b) Generating Kinoform, (c) Generating binary phase, (d) Exporting FZP pattern, (e) Performance verification

The comparison of the FZP pattern design performance between our proposal numerical simulation approach and the traditional analytical approach is important to demonstrate flexibility in this study. By step 4, some specific zones in radii (r_m) implemented by the two approaches described are compared in Table 4. The difference between whole designed zones is shown in Fig. 5(a) with a PV of 0.606 nm and goes further if the calculation resolution of increased. This sub-nm difference is dismissed compared to the most current advanced manufacturing technologies which are at the nanometer level. The evaluation is also carried out considering propagation at the focal plane (80 mm of displacement) of the FZP designed by both approaches. Fig. 5(b) shows a near-absolute agreement of the focal spot size in both, which has a confirmed FWHM = 17.55 μm equality.

Fig. 5 
Performance comparison of the FZP patterns designed by the numerical simulation and the analytical approach

The FZP pattern design by our numerical simulation is more verified by experiment. The practical FZP pattern is realized by photomasks technology and the fabrication positioning accuracy is used by 0.15 μm. A HeNe standard laser source (632.8 nm wavelength) is prepared from a fiber and collimated using a collimator. An expander, which is a pair of lenses in a Galilean configuration, is placed to increase the original beam size. A freespace beam from the collimator will pass through the expander and align with a displacement of 3 meters which guarantees plane wave characteristic within the experimental setup. This plane wave beam size is controlled by a precise iris and illuminates the FZP pattern as implemented in Fig. 6(a). A monochromatic detector with a resolution of 4,096 × 3,000 and pixel size of 3.45 μm is mounted on a translation stage to capture the beam shape along the light propagation y. The displacement accuracy of the translation stage is employed by 0.1 m and operated by a servo motor. The beam shapes at the different positions corresponding to the evaluation in ZEMAX-POP are taken and performed in Fig. 6(b). Beam propagation images are shown every 20 mm in each and a long displacement of 100 mm. At the focal plane in the experiment, the concentric faint sidelobes appear around the focal spot as a result of refraction of the light source obtained on the detector through the FZP pattern. They are considered dark current noise and negligible in this study. The number of data points taken by the detector is interpolated by MATLAB function to tens of thousands of data points to determine the FWHM of the focal spot. The focal spot size was recorded at FWHM = 18.55 μm and was also compared with the simulation result as shown in Fig. 7. The difference from the focal spot size in simulation by 1 μm that is the expected result based on manufacturing capability and alignment issues.

Fig. 6 
Practical FZP pattern verification by experiment

Fig. 7 
Focal spot size comparison between the numerical simulation and experiment