## Abstract

The effect of antireflection coatings on diffraction efficiency of diffractive optical elements (DOEs) was studied and the mathematical model of diffraction efficiency affected by antireflection coatings for DOEs is presented. We found antireflection coatings can cause a significant reduction on diffraction efficiency at the designed, or the central wavelength. In order to solve this problem, we proposed a method to keep 100% diffraction efficiency at the designed wavelength by ensuring the 2*π* phase induced by DOEs and the antireflection coatings. Diffraction efficiency affected by antireflection coatings for DOEs with consideration of antireflection coatings are simulated. Analysis results can be utilized for refractive-diffractive hybrid imaging optical system optimal design and image quality evaluation.

© 2017 Optical Society of America

## 1. Introduction

Diffractive optical elements (DOEs) are now commonly used in conjunction with different kinds of optical elements in numerous applications [1–3]. Diffraction efficiency is the key concern in applications [4]. The polychromatic integral diffraction efficiency(PIDE) stands for the illumination over the whole broadband, which can directly affect the modulation transform function of hybrid optical systems and future be used for the image quality evaluation of hybrid optical systems [5]. However, properties including diffraction efficiency and PIDE of DOEs are very sensitive to wavelength change. And the first-order diffraction efficiency can be maximized of 100% at the designed wavelengths.

Single-point turning technique has been widely used in DOEs of imaging optical systems manufacturing to achieve continuous surface relief. Materials includes both plastic and crystal. The optical elements, DOEs included, should be coated with antireflection coatings to increase transmittance of the refractive-diffractive hybrid optical system within the scope of working waveband. Antireflection coating of plastics for optical applications is intended to both mechanical durability of soft polymers improvement and antireflection function of optical systems [6]. Meanwhile, thin-film deposition technique has been used for antireflection coatings of DOEs [7,8].

In reference [9], inventor stated methods and apparatus to reduce reflection at the interface between a binary or multi-level diffractive elements and a surrounding medium. In reference [10], the authors analyzed the fabrication and characterization of a nano-structured diffractive element with near-zero reflection losses. In reference [11], authors investigated the superposition of sub-wavelength phase gratings onto blazed phase gratings to reduce surface reflections and increase diffraction efficiency. For traditional diffractive optical elements design, phase delay of antireflection coatings have not been taken into consideration. Wherever, antireflection coatings add optical path, which can be converged into phase of DOEs, which can reduce the diffraction efficiency at designed wavelengths and PIDE with the scope of waveband.

However, there has been no report, so far, of mathematical model for the effect caused by antireflection coatings and methods to solve this problem. In this paper, the mathematical model between diffraction efficiency as well as PIDE and antireflection coatings are presented and simulated. In addition, optimized method for diffraction efficiency reduction caused by antireflection coatings are proposed and analyzed. Example is discussed in visible waveband. Comparison of diffraction efficiency as well as PIDE with traditional and optimal methods for optical plastic material PMMA as DOEs substrate with antireflection coatings are simulated. Results can be used to guide optical engineers for hybrid optical system design.

## 2. Principle for optimal design method of DOEs with antireflection coatings

The DOEs have strong negative dispersion characteristics, which can be used for chromatic dispersion and high order aberrations correction of hybrid optical systems. Antireflection coatings on DOEs surface can add extra phase for DOEs micro-structure, further reduce diffraction efficiency at the designed wavelength and PIDE over the waveband. The model of antireflection coatings on the DOEs’ surface is shown in Fig. 1. Antireflection coatings are composed of multi-layers with different materials to achieve higher transmittance of optical systems. As shown in Fig. 1,${n}_{i}$stands for the index of coating material of each layer, ${l}_{i}$is antireflection coatings thicknesses of each layer. Model of DOEs with antireflection coatings is shown in Fig. 2. The solid part stands for the antireflection coatings. As is shown in Fig. 2, $H$stands for the micro-structure height of DOEs based on the traditional method design without consideration of phase caused by antireflection coatings.

As for imaging DOEs, the scalar theory is useful for designing surface relief diffractive elements with periods that are much larger than the wavelength for which the element is to be used. So, based on Fourier optics and scalar diffraction theory, the diffraction efficiency [4] of DOEs can be presented as

where $\mathrm{sin}c(x)=\raisebox{1ex}{$\mathrm{sin}(\pi x)$}\!\left/ \!\raisebox{-1ex}{$\pi x$}\right.$ and $\varphi (\lambda )$ the phase for DOEs.The phase for DOEs consist of DOEs body phase as well as antireflection coatings phase, which can be written as

where ${\varphi}_{DOEs}(\lambda )$ is the DOEs body phase and ${\varphi}_{AR}$ is the antireflection coatings phase.The expression for antireflection coatings phase can be expressed as

For traditional DOEs design, the micro-structure height of DOEs can be calculated according to [11]

where, $H$stands for the micro-structure height based on the traditional method design, meaning without antireflection coatings consideration. $n({\lambda}_{designed})$ stands for substrate index at the designed wavelength. ${n}_{0}({\lambda}_{designed})$ stands for the index of medium of substrate at the designed wavelength.Antireflection coatings can add phase to DOEs, which can lead to a reduction of diffraction efficiency further affect the image quality of hybrid optical systems. Also, the phase can be converged into micro-structure height, presenting as

Taking the phase of antireflection coatings into account and ensuring 100% diffraction efficiency at the designed wavelength, the optimal micro-structure height of DOEs should be derived as

Then, diffraction efficiency and PIDE with antireflection coatings can be expressed separately as Eqs. (9) and (10)

On optical transfer function related by diffraction efficiency of diffractive optical elements, DaleA. Buralli presented the influence of diffraction efficiency comparatively [5]. It can be seen actual modulation transfer function MTF values of optical systems can be approximated as the product between bandwidth PIDE and the theory OTF values, which can be expressed as

where, ${f}_{x},{f}_{y}$stand for the frequency of hybrid optical systems. Therefore, analyses results consisting of antireflection coatings can be used as evaluation of real image quality in refractive-diffractive hybrid optical system.## 3. Example and discussion

According to the previous analysis, for substrates PMMA as positive components, the micro-structure height for them can be calculated as 1.11um when the designed wavelength is chosen as 0.55um. Then, taking plastic optical materials PMMA as DOEs substrate, antireflection coatings for them are designed and optimized [6]. The antireflection coatings results for PMMA as substrate is shown in Table 1.

Form Eqs. (3) and (4), the phase of antireflection coatings can be calculated. And then the micro-structure height for DOEs can be calculated according to Eqs. (6) and (7).

Under traditional design method for DOEs, the relationship between diffraction efficiency and incident wavelength is simulated with MATLAB software, which is shown in Fig. 3. Diffraction efficiency in separable wavelengths is as shown in Table 2.

The dashed curve stands for the real diffraction efficiency of DOEs with antireflection coatings and the solid curve stands for the diffraction efficiency of DOEs without antireflection coatings.

From Fig. 3. and Table 2, we can see when taking PMMA as substrate, diffraction efficiency can achieve 100% at the designed wavelength. And when coated, the 100% diffraction efficiency exists towards the long wavelength. With antireflection coatings, the diffraction efficiency deduces from 100% to 97.8791% (a reduction of 2.1209%), which leads a reduction for MTF in hybrid optical system. In order to make sure 100% diffraction efficiency at designed wavelength, the DOEs structure parameters should be optimized. According to above theory and Eqs. (5)-(7), the micro-structure height can be calculated. The optimized diffraction efficiency can be simulated with MATLAB software. Result is shown in Fig. 4. It can be seen the diffraction efficiency at designed wavelength achieve 100% with antireflection coatings by the mean of optimal method (an increase of 2.1209% at designed wavelength).

From Table 3, we can see antireflection coatings can reduce the PIDE. Taking optical plastic PMMA as substrate, the PIDE without coatings is 88.4946% and the PIDE with antireflection coatings is 76.1501% (a reduction is 12.3445%). However, with the optimal method, the PIDE value is 84.8258%. Last but not least, the MTF of refractive-diffractive hybrid optical system can be increased with optimal method for PIDE.

## 4. Conclusions

In this paper, the mathematical model for effect on diffraction efficiency caused by antireflection coatings for DOEs and method of countermeasure for diffraction efficiency reduction caused by antireflection coatings have been established for the first time. The relationship between diffraction efficiency and antireflection coatings is analyzed which can be used for image evaluation of hybrid optical system with DOEs. Taking optical plastic materials commonly used PMMA as substrate for DOEs, diffraction efficiency and PIDE for DOEs are simulated with MATLAB software. The diffraction efficiency and PIDE reduce resulting from the antireflection coating, which will also lead to a reduction of MTF of refractive-diffractive hybrid optical system. In addition, with optimal method, the diffraction efficiency for designed wavelength can be achieved 100% and PIDE can be improved, which can benefit hybrid optical system design. In conclusion, the optimal method results can be taken into consideration to perfect DOEs during refractive-diffractive hybrid optical system design.

## Funding.

China Government (51-H34D01-8358-13/16).

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