## Abstract

Oblique incidence is the normal working mode for diffractive optical elements (DOEs). The diffraction efficiency is very sensitive to the angle of incidence for multilayer diffractive optical elements (MLDOEs). Therefore, the design and diffraction efficiency analysis of MLDOEs with wide angles of incidence is of universal significance and practice. We propose an integral diffraction efficiency model for MLDOEs with wide angles of incidence in case of polychromatic light and then describe this corresponding optimal design in detail. It is shown that high diffraction efficiency can be realized by the surface micro-structure heights optimization, ensuring high modulation transfer function (MTF) for MLDOEs with wide angles of incidence in hybrid optical systems. On this basis, we present the optimal design process and simulation of an MLDOE working in visible waveband with optical plastic materials combination PMMA and POLYCARB as the two-layer substrates. The result shows that with this optimal design, we can achieve the maximum diffraction efficiency and minimum micro-structure heights, which makes the MLDOE design exactly over the entire waveband and wide angles of incidence especially for zoom hybrid optical system.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Diffractive optical elements (DOEs) show the characteristics of both special dispersion and temperature [1]. It is of great significances for hybrid imaging optical system optimization, such as realizing a better image quality, miniaturizing the volume and weight, and avoiding the use of special optical materials. Diffractive-refractive hybrid imaging optical systems composed of DOEs and refractive lenses are widely used in varieties of high-end optical systems, militaries and other fields [2–5]. For DOEs, diffraction efficiency is the very important parameter that determines the working wavebands and applications. When the waveband and angle of incidence of a hybrid optical system becomes wider, the diffraction efficiency will be drastically reduced, thereby the imaging quality of hybrid optical system will be affected further.

Multilayer diffractive optical element (MLDOE) is a typical DOE, which can achieve a high diffraction efficiency over a wide waveband through a combination of different substrate materials. In Ref [6], D. A. Burali and G. M. Morris proved that the polychromatic integral diffraction efficiency (PIDE) directly affects the modulation transfer function (MTF) of the hybrid optical system. Therefore, design efforts should be made to improve the PIDE. In order to minimize the effects of the diffraction efficiency on image quality, the most common and effective method is to maximize the PIDE to achieve a high diffraction efficiency [7]. Optimal design of MLDOE with effective area method is given [8]. In addition, based on the point spread function (PSF) model, the image quality can be better [9].

An important feature of the MLDOE is its diffraction efficiency sensitivity to the angle of incidence, and the normal working mode of a MLDOE is just the oblique incidence. Moreover, effects on diffraction efficiency caused by the angle of incidence will lead to an inaccurate evaluation for MLDOE, resulting in the final image quality deduction for the hybrid imaging system. Therefore, how to achieve a high diffraction efficiency of MLDOE with wide angles of incidence is an inevitable problem that needs to be solved urgently in optimization design. There exist some ways to improve the diffraction efficiency reduction of a MLDOE caused by wide angles of incidence. First, a type of three-layer DOE is put forward to obtain high PIDE properties with wide angles of incidence [10–14]. Second, a high PIDE can be ensured by optical substrate material selection [15], whereas it is for a certain angle of incidence. However, what can be achieved by the existing manufacturing techniques is the double-layer DOE with air gap between them. So, how to achieve a high diffraction efficiency with wide angles of incidence is urgent to solve with a wider and more practical value, especially for zoom hybrid optical systems.

In this paper, we present a precious integral diffraction efficiency model and design for MLDOE with wide angles of incidence in polychromatic light. First, we propose the model of integral diffraction efficiency for MLDOE working with wide angles of incidence and wavebands. Then, we derive the corresponding mathematical expressions and present its optimal design for large angle of incidence in detail. This method based on integral diffraction efficiency with wide angles of incidence in case of polychromatic light can make the MLDOE design more accurate and reasonable. So, we can give a more precious design than that with the conventional designs, such as substrate optical material section and type of three-layer DOE. The benefit of this design is that the maximum integral diffraction efficiency overall the wide angles of incidence in case of polychromatic light for MLDOE can be achieved, thereby ensuring the high MTF of a hybrid imaging optical system. The MLDOE design taking the visible waveband with PMMA-POLYCARB as an example of the two substrates will demonstrate that this method and result can be applicable for hybrid imaging systems with wide angles of incidence such as the zoom hybrid optical system.

## 2. Integral diffraction efficiency model for MLDOEs with wide angles of incidence in case of polychromatic light

There are a number of works devoted to relevant studies in the framework of a rigorous diffraction theory for various types of MLDOEs working in different wavebands [16–19]. The scalar- and vector-diffraction theories are the two main theories for DOE design. For the optical-imaging system, since the feature size are several times larger than the incident wavelength, the scalar-diffraction theory can satisfy the design requirements and accuracies. And when the feature size is close to the incident wavelength, the strict vector diffraction theory should be used for MLDOE design and analysis.

For hybrid optical system, in order to maximize the diffraction efficiency of a MLDOE for the first diffraction order, the micro-structure heights for each layer should be of integer times of phase retardation at the design wavelengths (*λ _{1,} λ_{2。。。。。。}λ_{N}*). When the optical material combinations of a MLDOE are selected, the optical path differences (OPDs) corresponding to the design wavelength pairs and angles of incidence are expressed as [20]

*λ*represents the

_{k}*k*th design wavelength;

*H*indicates the height of the

_{j}*j*th micro-structure;

*n*

_{ji}(

*λ*) and

_{k}*n*

_{jt}(

*λ*) stand for the refractive indices of the incident and exit media of the

_{k}*j*th layer, respectively;

*θ*

_{ji}and

*θ*

_{jt}stand for the incident and the exit angles on the

*j*th layer, respectively;

*N*stands for the layer number and

*m*stands for the diffraction order. In addition, by solving Eq. (1) with a given combinations of angles of incidence and design wavelength pairs, the micro-structure heights of a MLDOE will result in a series of different values of (

*H*). Furthermore, the relationship between the OPD and its phase delay is

_{1,}H_{2。。。。。。}H_{N}*ϕ*(

*λ*,

*θ*) = (2π /

*λ*)OPD.

MLDOEs are with different types according to the micro-structure combinations, which all can achieve high diffraction efficiencies overall the whole waveband. One of the typical examples is a double-separated MLDOE composed of two different dispersive materials with air gap (*n _{m}*(

*λ*) = 1) between them, whose structure is shown in Fig. 1.

The diffraction efficiency of a MLDOE is very sensitive and gradually decreases to the angle of incidence, which is the key concern that cannot be ignored for its design. For the MLDOE used in hybrid imaging system, oblique incidence is the common working condition. Therefore, whether it is a system with a large angle of view or large aperture angle, the micro-structure heights should be optimally designed to achieve high diffraction efficiencies with wide angles of incidence in case of polychromatic light. As shown in Fig. 2, a MLDOE is applied in a hybrid imaging system with a wide angle of incidence, in which *α* and *β* stand for the large angle of view and angle of aperture.

Immediately note that the reliefs depths obtained within the framework of the scalar diffraction theory can serve as a good initial solution for optimization within the framework of the vector diffraction theory [21]. According to Fourier optics and scalar theory, the diffraction efficiency of a MLDOE with continuous surfaces is expressed as [22,23]

*η*(

_{m}*λ*,

*θ*) is the diffraction efficiency of the MLDOE, sinc(

*x*) = sin(π

*x*)/(π

*x*);

*ϕ*(

*λ*,

*θ*) is the phase delay, which is a function of wavelength

*λ*and angle of incidence

*θ*at a given micro-structure period, and when the phase delay is

*ϕ*(

*λ*,

*θ*) = 2π, the first-order diffraction efficiency can be obtained as 100%.

As shown in Fig. 2(b), when the light beam is obliquely incident onto the MLDOE substrate, the phase delay generated by the micro-structure heights, angle of incidence, waveband and material refractive properties can be expressed as

*H*and

_{1}*H*stand for the first- and second-layer micro-structure heights, respectively;

_{2}*θ*is the angle of incidence and

*λ*is the incident wavelength;

*n*

_{1}(

*λ*) and

*n*

_{2}(

*λ*) stand for the refractive indices of the first- and second-layer substrates, respectively.

Eventually, to ensure the 100% diffraction efficiency, the diffraction order should be equal to phase delay. From Eqs. (2) and (3), the micro-structure heights of the first- and second-layer should be calculated as

*A*and

*B*can be expressed as

By introducing Eqs. (3) and (4) into Eq. (2), the diffraction efficiency of the MLDOE can be obtained as [23]:

In order to consider the practical MLDOE applications comprehensively, the incident wavelength and angle should be both taken into consideration to maximize the diffraction efficiency. So, we propose an integral diffraction efficiency model for a MLDOE with wide angles of incidence in polychromatic light, which refers to the overall diffraction efficiency performance in consideration of both the incident waveband and angles of incidence. This relationship can be expressed as

*λ*

_{max}and

*λ*

_{min}represent the maximum and minimum wavelengths, respectively;

*θ*

_{max}and

*θ*

_{min}represent the maximum and minimum angles of incidence, respectively;

*λ*and

*θ*stand for the wavelength and angle of incidence, respectively.

The ultimate goal of MLDOE optimizing design is to obtain the maximum diffraction efficiency over the entire working angle of incidence and waveband, which is essentially a process of inversely obtaining the optimal solution. That is to say, according to the waveband and angle of incidence for the hybrid imaging system, the optimal design wavelength pair and micro-structure parameters are reversely obtained, thereby maximizing the diffraction efficiency. The optimization design process is mainly divided into three steps: the first step is to obtain the integral diffraction efficiency in case of polychromatic light (PIDE), the second step is to obtain its integral diffraction efficiency with wide angles based on the result of the first step, and the last step is the wavelength pair and micro-structure design of the MLDOE. The flow chart of this optimal design for MLDOE is shown in Fig. 3.

From Fig. 3, we can see that the integral diffraction efficiency with wide angles of incidence in case of polychromatic light is closely related to the design wavelength pair, incident angle range and micro-structure heights of MLDOE in a hybrid imaging system. Given a set of working condition (*θ, λ*_{1}, *λ*_{2}), the optimal micro-structure heights (*H*_{1}, *H*_{2}) can be calculated to obtain the maximum diffraction efficiency performance.

## 3. Example and analysis of the optimal design

According to this optimal model and design, as an example, a typical MLDOE with working waveband of 400 nm~700 nm, angle of incidence of *θ =* 0°~15°, and substrate materials combination of optical plastics PMMA-POLYCARB, is designed and analyzed. It is worth noting that, to achieve 100% diffraction efficiency, the micro-structure heights of (*H*_{1}, *H*_{2}) are different with the design wavelength pair and angle of incidence of (*θ, λ*_{1}, *λ*_{2}) selected.

From Eq. (5) and Fig. 3, the optimal micro-structure heights can be calculated when the angle of incidence and waveband are given. The micro-structure heights can be calculated when the first design wavelength is set in a certain step of *λ*_{1} + Δ*λ* in the waveband, and the second design wavelength is synchronized with the same step. Figure 4 shows the relationship between the micro-structure heights and design wavelength pair, where the horizontal coordinates (*x, y*) represent the two design wavelengths, respectively, and the *z*-coordinate is its surface micro-structure height.

Then, in the combinations of the two design wavelengths as a wavelength pairs, the integral diffraction efficiency with wide angles of incidence in case of polychromatic light can be obtained. The relationship between the design wavelength pair and integral diffraction efficiency with the working angle of incidence can be calculated, where the result is shown in Fig. 5. It shows that, when one of the design wavelengths is taken as a fixed value and the other design wavelength will take all the wavelengths along the y-coordinate axis, the corresponding series of integral diffraction efficiency with a wide angle of incidence in case of polychromatic light can be obtained at the same time.

More importantly, Fig. 5 also proves that the integral diffraction efficiency with wide angles of incidence in case of polychromatic light calculated by this optimal design is more accurate and reasonable for a real hybrid imaging system.

Next, the relationship between the first design wavelength and the integral diffraction efficiency with a wide angle of incidence can be obtained from Fig. 6. It shows that the maximum integral diffraction efficiency with a wide angle of incidence and the corresponding design wavelength pair can be obtained. More importantly, the micro-structure heights of a MLDOE can be achieved from Eq. (6). So, the corresponding maximum integral diffraction efficiency is 98.806% with the angle of incidence of 0°~15° and waveband of 400 nm~700 nm. When the corresponding optimal design wavelength pairs *λ*_{1} and *λ*_{2} are 450 nm and 550 nm, the micro-structure heights can be calculated as 15.364 μm and −11.929 μm, respectively.

The followings, we present this optimal design with the two typical design methods, one is the PIDE-based method (the design wavelength pair is of 435 nm and 598 nm) and the other is based on characteristic wavelength method (the design wavelength pair is 486nm and 656 nm). Following, we analyze the influences caused by angle of incidence and waveband on integral diffraction efficiency of a MLDOE.

#### 3.1. Influences of the angle of incidence

Figure 7 shows the results of the relationship between the diffraction efficiency and the angle of incidence for different design wavelength pairs, where the black, red, and green curves correspond to the first-order diffraction efficiencies of MLDOE at the design wavelength pairs of 450 nm and 550 nm, 435 nm and 598 nm, and 486 nm and 656 nm, respectively. It is shown that the diffraction efficiency of a MLDOE designed based on integral diffraction efficiency model with wide angles of incidence in case of polychromatic light is better than the other two methods within the wide angle of incidence.

Table 1 gives the calculated micro-structure heights and corresponding integral diffraction efficiencies at different design wavelength pairs within the incident angle of 0°~15°.

It can be seen that, considering the angle of incidence in range of 0°~15° and at the respective design wavelength pairs, the integral diffraction efficiency of MLDOE based on integral diffraction efficiency model with wide angles of incidence in case of polychromatic light is the highest. In addition, it can also minimize the micro-structure heights and maximize the diffraction efficiency within the incident angle range, which is higher than the other two methods with the values of 0.088% and 2.252%, respectively.

#### 3.2. Influences of the wavelength

Figure 8 shows the relationship between the diffraction efficiency of MLDOE and wavelength under different conditions of wavelength pairs, where the black, red, and green curves correspond to the design wavelength pairs of 450 nm and 550 nm, 435 nm and 598 nm, and 486 nm and 656nm, respectively. It is shown that the diffraction efficiency of MLDOE designed by using the integral diffraction efficiency model with wide angles of incidence in case of polychromatic light is better than the other two methods.

Table 2 gives the calculated micro-structure heights, diffraction efficiency and integral diffraction efficiency within the waveband of 400 nm~700 nm.

From Table 2, it can be seen that, integral diffraction efficiency is closely related to the angle of incidence. And with three designs comparisons, both the minimum diffraction efficiency and integral diffraction efficiency within the whole waveband based on this optimal design are much better than the other two typical designs. When considering the angle of incidence is 15°, at the respective design wavelength pairs, the integral diffraction efficiency of the MLDOE reaches the highest based on the integral diffraction efficiency model with wide angles of incidence in polychromatic light, reaching 97.707%, higher than the other method with the values of 1.943% and 3.783%, respectively. In addition, the minimum diffraction efficiency of the MLDOE reaches the highest based on the integral diffraction efficiency model with wide angles of incidence in polychromatic light, reaching 94.435%, higher than the other method with the values of 3.567% and 13.556%, respectively.

#### 3.3. Influences of the incident angle and wavelength

Figure 9 shows the influences of both the incident angle and wavelength on the diffraction efficiency of MLDOE with different design wavelength pairs. Figures 9(a)–9(c) show the relationships among diffraction efficiency, wavelength and incident angle on MLDOE based on integral diffraction efficiency model with wide angles of incidence, PIDE, and characteristic wavelength method, whose wavelength pairs correspond to 450 nm and 550 nm, 435 nm and 598 nm, 486 nm and 656 nm, respectively.

Table 3 gives the calculated micro-structure heights and integral diffraction efficiency within the waveband of 400 nm~700 nm and the angle of incidence of 0°~15°.

It can be seen that, considering the angle of incidence and the design wavelength pairs, the integral diffraction efficiency can be the maximum, reaching 98.806%, higher than the other method with the values of 0.357% and 2.768%, respectively. Last but not least, the micro-structure heights based on integral diffraction efficiency model can be the minimum, corresponding to 15.364 μm and −11.929 μm, which can be manufactured with the minimum manufacturing errors and further ensure the minimum MTF reduction of a hybrid optical system. Most importantly, for the processing capability of Nanoform 700 ultra, whose manufacturing capacity of its surface roughness is less than 1 nm. So, it is easy for this DOE manufacturing in accordance with the design surface structure at a high precision.

## 4. Conclusion

Considering the diffraction efficiency is sensitive to the incident angle, and it can cause MTF reduction of a real hybrid optical system. The current designs are from aspects such as new structure, material selection, certain angle, etc. Thus, we propose an integral diffraction efficiency model for MLDOEs with wide angles of incidence in case of polychromatic light and present this optimal design in detail. With this method, the maximum integral diffraction efficiency can be calculated by the design wavelength pair selected, and in addition, the minimum micro-structure heights can be obtained with wide angles of incidence in case of polychromatic light for MLDOEs. The result shows that the MLDOEs based on this optimal model and design is adopted, which is superior to the other methods. This integral diffraction efficiency model with wide angles of incidence in case of polychromatic light is more practical and the optimization design is of better performance. All in all, what we believe this is a new method of MLDOEs optimal design for improving diffraction efficiency within the whole angle of incidence and broad waveband. The optimization design of MLDOEs based on this optimal model can be of great significance and more reasonable for the quantization and optimization in the practically engineered hybrid imaging optical systems.

## Funding

National Natural Science Foundation of China (NSFC) (11634010); China Postdoctoral Fund (2018M643728); Natural Science Foundation of Shannxi Province (2019JQ-063); Fundamental Research Funds for the Central Universities (310201911CX045).

## References

**1. **G. J. Swanson, “The theory and design of multi-level diffractive optical elements,” MIT Lincoln Laboratory technical report (1989).

**2. **B. Sabushimike, G. Horugavye, and S. Habraken, “Optimization of a multiblaze grating in reflection using a free-form profile,” Appl. Opt. **57**(18), 5048–5056 (2018). [CrossRef] [PubMed]

**3. **O. Barlev and M. A. Golub, “Multifunctional binary diffractive optical elements for structured light projectors,” Opt. Express **26**(16), 21092–21107 (2018). [CrossRef] [PubMed]

**4. **C. Wu, H. Gu, Z. Zhou, and Q. Tan, “Design of diffractive optical elements for subdiffraction spot arrays with high light efficiency,” Appl. Opt. **56**(31), 8816–8821 (2017). [CrossRef] [PubMed]

**5. **C. Bigwood and A. Wood, “Two-element lenses for military applications,” Opt. Eng. **50**(12), 121705 (2011). [CrossRef]

**6. **D. A. Buralli and G. M. Morris, “Effects of diffraction efficiency on the modulation transfer function of diffractive lenses,” Appl. Opt. **31**(22), 4389–4396 (1992). [CrossRef] [PubMed]

**7. **C. Xue and Q. Cui, “Design of multilayer diffractive optical elements with polychromatic integral diffraction efficiency,” Opt. Lett. **35**(7), 986–988 (2010). [CrossRef] [PubMed]

**8. **H. Yang, C. Xue, C. Li, J. Wang, and R. Zhang, “Optimal design of multilayer diffractive optical elements with effective area method,” Appl. Opt. **55**(25), 7126–7133 (2018). [CrossRef] [PubMed]

**9. **Y. Hu, Q. Cui, L. Zhao, and M. Piao, “PSF model for diffractive optical elements with improved imaging performance in dual-waveband infrared systems,” Opt. Express **26**(21), 26845–26857 (2018). [CrossRef] [PubMed]

**10. **Y. H. Zhao, C. J. Fan, C. F. Ying, and S. H. Liu, “The investigation of triple-layer diffraction optical element with wide field of view and high diffraction efficiency,” Opt. Commun. **295**, 104–107 (2013). [CrossRef]

**11. **S. Mao, Q. Cui, M. Piao, and L. Zhao, “High diffraction efficiency of three-layer diffractive optics designed for wide temperature range and large incident angle,” Appl. Opt. **55**(13), 3549–3554 (2016). [CrossRef] [PubMed]

**12. **C. Fan, “The investigation of large field of view eyepiece with multilayer diffractive optical element,” Proc. SPIE **9272**, 92720N (2014). [CrossRef]

**13. **Y. Zhao, C. Fan, C. Ying, and H. Wang, “The investigation of three layers diffraction optical element with wide field of view and high diffraction efficiency,” Optik (Stuttg.) **124**(20), 4142–4144 (2013). [CrossRef]

**14. **H. Xie, D. Ren, C. Wang, C. Mao, and L. Yang, “Design of high-efficiency diffractive optical elements towards ultrafast mid-infrared time stretched imaging and spectroscopy,” J. Mod. Opt. **65**(3), 255–261 (2018). [CrossRef]

**15. **B. Zhang, Q. Cui, and M. Piao, “Effect of substrate material selection on polychromatic integral diffraction efficiency for multi-layer diffractive optics in oblique incident situation,” Opt. Commun. **415**(15), 156–163 (2018). [CrossRef]

**16. **F. Huo, W. Wang, and C. Xue, “Limits of scalar diffraction theory for multilayer diffractive optical elements,” Optik (Stuttg.) **127**(14), 5688–5694 (2016). [CrossRef]

**17. **G. I. Greĭsukh, E. G. Ezhov, S. A. Stepanov, V. A. Danilov, and B. A. Usievich, “Spectral and angular dependences of the efficiency of diffraction lenses with a dual-relief and two-layer microstructure,” J. Opt. Technol. **82**(5), 308–311 (2015). [CrossRef]

**18. **G. I. Greisukh, V. A. Danilov, E. G. Ezhov, S. A. Stepanov, and B. A. Usievich, “Spectral and angular dependences of the efficiency of relief-phase Diffractive Lenses with Two- and Three-Layer Microstructures,” Opt. Spectrosc. **118**(6), 964–970 (2015). [CrossRef]

**19. **G. I. Greisukh, V. A. Danilov, S. A. Stepanov, A. I. Antonov, and B. A. Usievich, “Spectral and Angular Dependences of the Efficiency of Three-Layer Relief-Phase Diffraction Elements of the IR Range,” Opt. Spectrosc. **125**(1), 60–64 (2018). [CrossRef]

**20. **D. C. O. Shea, T. J. Suleski, A. D. Kathman, and D. W. Prather, *Diffractive Optics Design, Fabrication, and Test* (SPIE, 2004).

**21. **G. I. Greisukh, V. A. Danilov, S. A. Stepanov, A. I. Antonov, and B. A. Usievich, “Minimization of the Total Depth of Internal Saw-Tooth Reliefs of a Two-Layer Relief-Phase Diffraction Microstructure,” Opt. Spectrosc. **124**(1), 98–102 (2018). [CrossRef]

**22. **A. P. Wood and P. J. Rogers, “Diffractive optics in modern optical engineering,” Proc. SPIE **5865**, 58650B (2005). [CrossRef]

**23. **Y. Arieli, S. Ozeri, N. Eisenberg, and S. Noach, “Design of a diffractive optical element for wide spectral bandwidth,” Opt. Lett. **23**(11), 823–824 (1998). [CrossRef] [PubMed]