Implementation and Verification of Fresnel Zone Plate Patterns Designed by Optimization of Surface Phase

1. Introduction

The trend of shrinking and thinning optical systems has always been a priority in the design of optical devices. In recent years, flat lenses have emerged as leading candidates to realize this trend [1-3]. Unlike conventional bulk lenses that use refractive theory to modulate the phase and control light propagation, Fresnel Zone Plates (FZP) is one of the most basic flat lenses that utilize diffraction theory to control waves [4,5]. They serve as the basis for modifying various diffractive patterns to address specific optical issues, such as depth of focus (DOF) extension, freeform wavefront metrology, compact imaging systems, and aberration correction [6-9]. They are widely applied in microscopy, astronomy, and telecommunications due to their thin structure.

The fundamental principle of FZP is established in dividing concentric rings into alternating transparent and opaque zones to generate out-of-phase π depicted in Fig. 1(a), allowing them to superpose diffraction rays as foci [10]. Most of the FZP is designed through an analytical representative of the exact size of zones r_m² +f² = (f + mλ)², where r_m is a radius of m^th ring zone (m = 1,2, 3...), f and f_m is the foci position of FZPs and m^th zones, respectively, λ is the designed wavelength, and f_m = f + mλ. Then, the light propagation is considered based on Rayleigh-Sommerfeld diffraction theory as Fig. 1(b) [11].

Fig. 1

Fresnel zone plate is designed by the analytical method

Indeed, with the rapid advancement of photolithography and laser writer technique, the FZP is a high-resolution solution for an optical promise, but its major drawback is low efficiency [12-14]. Various ideas have been proposed to improve this issue, such as kinoform, binary phase plate, and refractive-diffractive hybrid optics [15-18]. However, the analytical design approach is quite rigid and inflexible, making it challenging to vary the size of the zones to achieve the desired combined performance.

In this study, we introduce a numerical approach for designing the FZP patterns, that can be easily modified through an optimization of surface phase using optical ray-tracing software [19]. The flexible numerical approach allows the FZP pattern to exhibit multifunctionality and adapt to various applications. We will discuss the detailed design procedure, evaluate performance, conduct a comparison with the analytical solution, and then validate it through experimentation.

2. Design Method

We introduce a general design procedure for the FZP pattern in Fig. 2(a) and their details in Fig. 2(b) include 5 main steps.

Fig. 2

Fresnel zone plate pattern designed procedure

2.2 Numerical Design and Simulation

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.

Table 1

Design specifications

Fig. 3

Numerical simulation for the FZP pattern

Table 2

Merit function editor (MFE)

Table 3

Binary Optic 2 surface phase specification

2.3 Performance

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

3. Verification

3.1 Comparison

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.

Table 4

Zone comparison in radii

Fig. 5

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

3.2 Experiment

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

4. Conclusion

In this study, we present a flexible procedure to design the FZP pattern using the numerical simulation approach. By optimizing the surface phase with the desired target operands in ZEMAX^® software, the extracted surface phase is converted to the FZP pattern. Our approach is demonstrated by a near-absolute agreement of the focal spot size of 17.549 μm and the sub-nm difference of the zones in radii with the analytical approach. Furthermore, it was also verified by the practical experiment, and the focal spot size was expected as 18.55 μm. There have been many attempts to simulate various functionality FZP patterns and combine refractive-diffractive hybrid lenses to solve optical issues such as DOF modulation, aberration correction, or freeform wavefront metrology. Our approach has the potential to be applied in the above combination design and can be freely modified depending on optimization targets that cannot be met by the traditional analytical approach. Besides, a high-precision FZP pattern design is implemented that far exceeds the fabrication capabilities currently available with typical technologies.

Acknowledgments

This study was supported by the National Research Foundation with the government, Ministry of Science and ICT (No. 2021R1A4A1031660), “Fanout semiconductor PR pattern highspeed automatic inspection equipment development (No. 20018441)”, a part of the Ministry of Trade, Industry and Energy's mechanical equipment industrial technology development project, and “Development of Ultra-Small Satellite with Hyperspectral and Visible Dual-Sensor Camera (No. RS-2022-00155901)”, a part of the space innovation project funded by the National Research Foundation of Korea.

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