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Diffractive optical element optimization under wide incident angle and waveband situations
Optics Communications 458 (2020) 124762
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Diffractive optical element optimization under wide incident angle and waveband situations Shan Mao, Jianlin Zhao ∗ MOE Key Laboratory of Material Physics and Chemistry under Extraordinary Conditions, and Shaanxi Key Laboratory of Optical Information Technology, School of Science, Northwestern Polytechnical University, Xi’an 710072, China
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Keywords: Diffractive optical element Diffraction efficiency Incident angle range Optimal design
ABSTRACT The single-layer diffractive optical elements (SLDOEs) exhibit various advantages with respect to the design, configuration, fabrication, and cost. However, they also rapidly decrease the diffraction efficiency in a wide working waveband and incident angle range. In this study, we propose an optimal design and set up a mathematical model and derive expressions for SLDOE in actual applications to effectively solve this problem. Initially, an optimal SLDOE design can be obtained by establishing the mathematical model and diffraction efficiency expressions for SLDOEs with respect to the incident angle. Finally, the diffraction efficiency within an incident angle range is presented based on three typical designs, and their influences are analyzed. The results denote that the angle bandwidth integrated average diffraction efficiency (ABIAED)-based design can improve the diffraction efficiency over the entire waveband and range of incident angles, while improving the ease of processing of SLDOEs with a minimum microstructure height.
1. Introduction Diffractive optical elements (DOEs) exhibit special dispersion and thermal properties that considerably enhance their usefulness in several applications [1,2]. When size or weight of the optical system is crucial, DOEs provide substantial advantages and can be used to design an aberration-minimizing compact imaging system thinner and lighter than a system manufactured using traditional reflective or refractive optical elements. DOEs can provide additional design freedom for various electro-optical instrument applications, such as while being used in conjunction with other optical components or producing desired effects on their own depending on what the optical designers are attempting to achieve for special design concepts [3–5]. The commonly used hybrid optical lens comprises a conventional refractive or reflective element as the substrate combined with a DOE micro-structure on one surface. In this system, DOEs can be used to correct image aberrations, especially color aberrations, considerably similar to aspheric surfaces or additional refractive lenses [6]. Thus, they can be extensively applied to visible and infrared cameras for aberration correction in militaries, high-end optical systems, and other such applications [7–9]. Current research progresses have made it possible to design and manufacture DOEs with high micro-structure precision [10] and diffraction efficiency. Additionally, many studies related to the DOE implementation methods and performance evaluation have been published [11–15]. Currently, the effective energy imaged using DOE can be attributed to a first-order diffractive beam. For the remaining orders, ∗
the beam is diffused throughout the image plane, resulting in the creation of harmful energy. To address this problem, the detector uses a point spread function (PSF) model for image collection, and high diffractive order energies converge to the image plane. The DOE image quality will be substantially improved when the effective energy of the system is combined with the design method and the PSF image processing model [16]. Typical types of DOE include single-layer DOEs (SLDOE), multilayer DOEs (MLDOE), and harmonic DOEs (HDOE). SLDOE is the easiest to design and manufacture; however, SLDOEs exhibit a significant limitation with respect to the strong diffraction efficiency sensitivity caused by the wavelength derived from the design wavelength and working incident angle. Diffraction efficiency is a major concern for optical engineers; therefore, effective designs should be established to improve the diffraction efficiency. Recent research has resulted in some excellent SLDOE designs based on light being incident perpendicularly or under a certain incident angle to the SLDOE surface. Obviously, all these designs do not consider the influence of the incident angle ranges on the phase delay and diffraction efficiency. Two important designs are obtained based on the mid-wavelength and bandwidth integrated average diffraction efficiency (BIADE) solutions [17]. The SLDOEs applied in hybrid optical systems are always under oblique incidence conditions on its surface, indicating that the design optimization and diffraction efficiency analysis of SLDOEs are universally significant and practical under oblique
Corresponding author. E-mail address: [email protected] (S. Mao).
https://doi.org/10.1016/j.optcom.2019.124762 Received 6 August 2019; Received in revised form 8 October 2019; Accepted 13 October 2019 Available online 17 October 2019 0030-4018/© 2019 Elsevier B.V. All rights reserved.
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Optics Communications 458 (2020) 124762
Fig. 1. The incident angle distributions on a single-layer diffractive optical element surface for (a) a large aperture system and (b) a large field-of-view system.
incidence conditions. This is the case even for a large view angle or aperture angle, which is practical, especially in zoom hybrid optical systems typically applied to high-end optical systems. Therefore, the major objective of this study is to present an optimal SLDOE design and the related diffraction efficiency optimization using design wavelength selection and DOE parameter calculations. In this study, we propose a concept for angle bandwidth integrated average diffraction efficiency (ABIADE) and establish a corresponding mathematical model. Subsequently, we performed SLDOE optimization based on the aforementioned concept and model. The optimized design can ensure high diffraction properties for a SLDOE over a wide range of incident angle, ensuring the imaging quality of a hybrid optical system. Furthermore, we explored a novel method for evaluating SLDOEs in a broad waveband range with large incident angles. This method can expand the working waveband range when compared with that in the traditional designs. In addition, optical engineers can apply the analysis described in this study to both the MLDOE and HDOE lenses.
the SLDOE diffraction efficiency based on the mid-wavelength design at several incident wavelengths. When the long-infrared waveband is selected as the working waveband and 0◦ ∼45◦ is selected as the incident angle range, the diffraction efficiency can be evaluated; the results are presented in Fig. 2. Fig. 2(a) shows the relation among the incident angle, wavelength, and corresponding diffraction efficiency, Fig. 2(b) shows the relation between the incident angle and overall BIADE with respect to the long-infrared band, and Fig. 2(c) shows the relation between the wavelength and diffraction efficiency under different incident situations. As can be observed, Fig. 2(b) and (c) are derived from Fig. 2(a). Fig. 2 denotes that when the incident angle gradually increases, the diffraction efficiency initially decreases slowly, indicating that the SLDOE can be used in this angle range. However, the diffraction efficiency drastically declines as the incident angle increases further. Therefore, the incident angle formed by the incident light and SLDOE surface decreases the diffraction efficiency to the decreased diffraction efficiency obtained by design wavelength deviation. Hence, the decrease in the incident-angle-driven efficiency is an important problem that cannot be ignored in current SLDOE designs.
2. Problem statement Diffraction efficiency is an important parameter that determines the working waveband and DOE application. The modulation transform function (MTF) is directly affected by the BIADE for a hybrid optical system or lens [18]. The diffraction efficiency drastically decreases when the conventional DOE’s working waveband becomes wider and the incident angle increases, reducing the overall BIADE waveband and the MTF of the hybrid optical system. Therefore, the high diffraction efficiency requirements can be satisfied by reselecting the design wavelengths for achieving microstructure optimization. In hybrid imaging optical systems, a certain angle of light is incident on the SLDOE surface when the light is incident on the SLDOE itself and when the SLDOE works within a certain incident angle range in a hybrid optical system. Fig. 1 illustrates a hybrid imaging lens using a SLDOE, with Fig. 1(a) denoting the incident angle of a large aperture and Fig. 1(b) denoting the incident angle of a large field of view. As depicted in Fig. 1, the normal incident angle of a beam on the SLDOE surface is one of the simplest parameters. Generally, the beam transmission and diffraction efficiency at a certain incident angle should be considered. Therefore, an optical system is always with a large angle of view or a large aperture angle; then, the diffraction efficiency and the BIADE analysis of the SLDOE in an oblique incident angle range exhibits universal significance and is considerably practical. Fig. 2 shows the influence curves of the incident angle on
3. Optimization design The analyses presented in Section 2 demonstrate that the incident angle must be considered in SLDOE optimization to satisfy the high diffraction efficiency requirements within an incident angle range to ensure high imaging quality for hybrid imaging optical systems. A set of values should be obtained with respect to all the working wavelength and incident angle pair values for an SLDOE, i.e., (𝜆1, 𝜃1 ), (𝜆2, 𝜃2 ), (𝜆3, 𝜃3 ), … , (𝜆𝑖, 𝜃𝜄 ), to achieve high diffraction efficiency. Thus, by considering different working wavelength and incident angle pairs during the calculation, maximum diffraction efficiency can be obtained for a certain SLDOE. Subsequently, the microstructure height can be calculated based on the unique optimal design wavelength to achieve maximum BIADE [1]. SLDOEs achieve an optimal design using an ABIADE-based design at a certain incident angle and waveband. Further, the optimal design wavelength and diffractive microstructure parameters can be obtained based on the working waveband and incident angle range of a hybrid optical system, optimizing the maximum ABIADE value. In this section, SLDOE is optimized to achieve maximum ABIADE, where the optimal wavelength and microstructure height are considered to be the intermediate values. The process of maximizing ABIADE can be divided into the 2
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Optics Communications 458 (2020) 124762
Fig. 2. Wavelength and incident angle diffraction efficiencies. (a) Diffraction efficiency that can be attributed to the wavelength and incident angle; (b) angle bandwidth integrated average diffraction efficiency that can be attributed to the incident angle; and (c) the diffraction efficiency for different wavelengths at different incident angles.
following three parts: determining the maximum ABIADE; optimizing the design wavelength; and calculating the micro-structure height. Because the incident angle range has been presented at the beginning of the optical system design, we do not optimize the incident angle in this section. Fig. 3 depicts the SLDOE optimization design process. In Fig. 3, 𝜂3 represents the ABIADE maximum value of 𝜂 𝑚 (𝜆, 𝜃), where 𝜆 denotes the design wavelength and 𝜃 denotes the incident angle. 𝜂1 and 𝜂2 denote the diffraction efficiencies over all the incident angles and waveband. Further, 𝜆max represents the maximum and minimum wavelengths, and 𝜃max and 𝜃min represent the maximum and minimum incident angles, respectively. The specific optimization design idea is that the ABIADE can be calculated point–point with the angle range waveband considering the incident angle range and waveband; additionally, the ABIADE values corresponding to different incident angles and wavelengths is compared to obtain a one-to-one correspondence for the maximum ABIADE. Finally, the design wavelength can be determined. According to this design wavelength, the diffraction microstructure height of the SLDOE can be calculated; thus, the optimal SLDOE design can be achieved. Based on Fourier optics and scalar approximation theory, we present the optimization analysis of SLDOE. When we do not consider material absorption, reflection, and scattering of incident light, the diffraction efficiency of a continuous surface can be expressed as [1] [ ]2 𝜂𝑚 = sinc(𝑚 − 𝜙(𝜆, 𝜃)) , (1)
Fig. 3. The optimal flow for a single-layer diffractive optical element.
where sinc(𝑥) = sin(𝜋𝑥)∕(𝜋𝑥), 𝜙(𝜆, 𝜃) is the phase delay, 𝜃 is the incident angle, and 𝜆 represents the wavelength. m denotes the diffraction order, and only the first diffraction order of 𝑚 = 1 is usually considered to denote the effective energy. When determining the SLDOE material, incident angle 𝜃, and wavelength 𝜆, the phase delay should be equal to the diffraction order to
ensure 100% diffraction efficiency. Therefore, the corresponding phase delay from Eq. (1) can be expressed as [19] [√ ] 𝐻 𝜙(𝜆, 𝜃) = 𝑛2 (𝜆) − 𝑛2𝑚 (𝜆) sin2 𝜃 − 𝑛𝑚 (𝜆) cos 𝜃 , (2) 𝜆 3
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Optics Communications 458 (2020) 124762
where H is the micro-structure height, n(𝜆) is the refractive index of the substrate material, and 𝑛𝑚 (𝜆) is the refractive index of the medium. Thus, by substituting Eq. (2) into Eq. (1), the diffraction efficiency of SLDOE can be expressed as { [√ ]} 𝐻 𝑛2 (𝜆) − 𝑛2𝑚 (𝜆) sin2 𝜃 − 𝑛𝑚 (𝜆) cos 𝜃 . (3) 𝜂𝑚 (𝜆, 𝜃) = sinc2 1 − 𝜆 To maximize the first-order diffraction efficiency of a hybrid tical system, the phase delay because of its phase height should come an integral multiple of the design wavelength. Subsequently, microstructure height can be obtained as [√ ] 𝐻 𝑛2 (𝜆0 ) − 𝑛2𝑚 (𝜆0 ) sin2 𝜃 − 𝑛𝑚 (𝜆0 ) cos 𝜃 = 𝜆0 .
opbethe
(4)
When the design wavelength 𝜆0 exhibits different values in the working waveband and when the incident angle 𝜃 is any value within the angle range, the micro-structure height H will lead to a series result. The important indicators for designing and evaluating the SLDOE should consider factors, such as the working waveband, because the diffraction efficiency at a single wavelength cannot denote the overall performance. Additionally, the incident angle of light on the microstructure surface has been included in the original design specifications. Therefore, an ABIADE concept has been proposed in this study. The incident light in an optical system is generally not from a monochromatic source; therefore, it is meaningful to evaluate the BIADE that affects system imaging. In case of SLDOE, ABIADE refers to the overall imaging performance with respect to the incident angle range and working waveband, which can be expressed as 𝜂 𝑚 (𝜆, 𝜃) = =
𝜃
𝜆
𝜃
𝜆
max max 1 𝜂𝑚 𝑑𝜆𝑑𝜃 ∫ ∫ 𝜆max − 𝜆min 𝜃min 𝜆min
,
max max 1 sinc2 [1 − 𝜙(𝜆, 𝜃)] 𝑑𝜆𝑑𝜃 𝜆max − 𝜆min ∫𝜃min ∫𝜆min
Fig. 4. Relations between the refractive index and wavelength in different wavebands for several optical materials.
based on the BIADE requirements, ensuring the MTF of the system and yielding optimal image quality for the hybrid optical system under normal incidence. However, when the beam is incident onto the SLDOE at a certain angle, ABIADE is the corresponding value of the available waveband within a large incident angle range. Finally, the diffraction efficiency corresponding to different designs varies, and the differences in diffraction efficiency can be expressed as
(5)
⎧𝛥𝑠 = 𝜂 (𝜃1 , 𝜃2 , … 𝜃𝑖 ; 𝜆1 , 𝜆2 , … 𝜆𝑖 ) 𝐴𝐵𝐼𝐴𝐷𝐸 ⎪ 1 − 𝜂 𝑚𝑖𝑑−𝑤𝑎𝑣𝑒𝑙𝑒𝑛𝑔𝑡ℎ (𝜃1 , 𝜃2 , … 𝜃𝑖 ; 𝜆1 , 𝜆2 , … 𝜆𝑖 ) ⎪ . ⎨ ⎪𝛥𝑠 2 = 𝜂 𝐴𝐵𝐼𝐴𝐷𝐸 (𝜃1 , 𝜃2 , … 𝜃𝑖 ; 𝜆1 , 𝜆2 , … 𝜆𝑖 ) ⎪ − 𝜂 𝐵𝐼𝐴𝐷𝐸 (𝜃1 , 𝜃2 , … 𝜃𝑖 ; 𝜆1 , 𝜆2 , … 𝜆𝑖 ) ⎩
where 𝜆max and 𝜆min represent the maximum and minimum wavelengths, respectively, and 𝜃max and 𝜃min represent the maximum and minimum incident angles, respectively. In case of a conventional SLDOE, BIADE is observed to directly affect the imaging quality of the hybrid optical system. An optical imaging system’s modulation transfer function (𝑀𝑇 𝐹 𝑝𝑜𝑙𝑦 ) is the product of BIADE and can be expressed as [20] { 𝜂 ⋅ 𝑀𝑇 𝐹𝑖𝑑𝑒𝑎𝑙 (𝑓𝑥 , 𝑓𝑦 ), 𝑓𝑥 ≠ 0, 𝑓𝑦 ≠ 0 . (6) 𝑀𝑇 𝐹𝑝𝑜𝑙𝑦 = 1, 𝑓𝑥 = 𝑓𝑦 = 0
Generally, the design process of a hybrid optical system can be divided into the following two steps: the first step is the optimization design and calculations for the SLDOE and the second step is the optimization design for the hybrid optical system with respect to the SLDOE. 4. Simulation and analysis
In Eq. (6), 𝑀𝑇 𝐹 𝑖𝑑𝑒𝑎𝑙 denotes the ideal MTF of the hybrid optical system based on the optical design software and 𝑓𝑥 and 𝑓𝑦 denote the frequencies toward the x and y directions, respectively. Therefore, by considering the incident angle and substituting Eq. (5) into Eq. (6), the real MTF for a hybrid optical system can be expressed as { 𝜂 𝑚 (𝜆, 𝜃) ⋅ 𝑀𝑇 𝐹𝑖𝑑𝑒𝑎𝑙 (𝑓𝑥 , 𝑓𝑦 ), 𝑓𝑥 ≠ 0, 𝑓𝑦 ≠ 0 . (7) 𝑀𝑇 𝐹𝑝𝑜𝑙𝑦 = 1, 𝑓𝑥 = 𝑓𝑦 = 0
Optical materials, such as optical plastics and glasses in the visible waveband and crystalline materials in the infrared waveband, exhibit significantly different characteristics, resulting in large diffraction efficiency and micro-structure differences for SLDOEs. The optical handbook [21] states that the refractive indices and Abbe number of the infrared crystal materials are significantly larger than those of the optical plastics and glasses. Additionally, using infrared wavebands as an example, the optical performance for the same crystal material differs while working in the long-infrared (8∼12 μm) and mid-infrared wavebands (3∼5 μm). According to the optical software and material dispersion expressions, Fig. 4 simulates the relations between the refractive index and wavelength in different wavebands for several optical materials. Although optical plastics are not commonly used in highperformance optical lenses for hybrid optical systems, DOEs with high-precision continuous surfaces can be produced based on the development of the single-point diamond turning (SPDT) technique. In addition, the infrared crystal material has a larger Abbe number than those of glasses and optical plastics. When a diffractive–refractive hybrid lens satisfies achromatic and thermal conditions under infrared light, DOE exhibits a small distribution of optical power. Additionally, the number of diffraction cycle rings is significantly lower than that
When the light is perpendicularly incident to the SLDOE, based on Eq. (5), BIADE can be approximated from Eq. (5) as 𝜂 𝑚=1 ≈ 1 +
𝜋2 𝜋2 2 (𝜆min + 𝜆max − 𝜆0 ) − (𝜆 + 𝜆min 𝜆max + 𝜆2max ). 3𝜆0 9𝜆20 min
(8)
When the design wavelength 𝜆0 is determined, the BIADE of SLDOE can also be determined; thus, the selection of design wavelength can directly affect the optical system’s MTF. On both sides of Eq. (8), the design wavelength 𝜆0 can be calculated based on as 𝜆0 =
2(𝜆2min + 𝜆max 𝜆min + 𝜆2max ) 3(𝜆max + 𝜆min )
.
(10)
(9)
When the design wavelength can satisfy Eq. (9), BIADE reaches its maximum and minimally affects the image quality. Further, the available working waveband of the optical system can be determined 4
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Optics Communications 458 (2020) 124762
Fig. 6. Relation between the angle bandwidth integrated average diffraction efficiency and wavelength within the whole incident angle range.
Fig. 5. Micro-structure height caused by the incident wavelength and incident angle.
4.2. Analysis
in DOEs based on optical glasses and plastics. Therefore, we select a crystalline material that can work in a long-infrared waveband for performing simulation and analysis. Based on the material characteristics and Eq. (9), the design wavelength based on two traditional methods and microstructure heights under normal and oblique incident angles can be calculated, and the results are presented in Table 1. As presented in Table 1, the micro-structures differ under two incident situations. The microstructure heights and their differences in case of the long-infrared waveband are much greater than those for the visible and middle-infrared SLDOE.
The comprehensive effects of incident angle and wavelength on the diffraction efficiency was analyzed for different design wavelengths (based on different methods shown in Fig. 7), as shown in Fig. 7. Fig. 7 shows the diffraction efficiency effects caused by the incident angle and wavelength for ABIADE-, mid-wavelength-, and BIADE-based designs with design wavelengths of 9.55, 10, and 10.13 μm, respectively. The comparisons in Fig. 7 show that the overall diffraction efficiency of the ABIADE design is higher than those of the BIADE or midwavelength designs and that the performance has been significantly improved. As presented in Table 1, in the working angle range of 0◦ ∼30◦ and a waveband of 8∼12 μm, the diffraction efficiency can reach its maximum with the ABIADE-based design, greater than those of the BIADE- and mid-wavelength-based designs.
4.1. Optimal design With respect to the pairs of wavelengths and incident angles, the different microstructure heights can be calculated based on Eqs. (3) and (4); thus, the diffraction efficiency of the SLDOE exhibits minimal influence on the MTF and image quality for a hybrid optical system. Based on the optimal design theory, we assume that the design wavelength is 𝜆0 , the incident angle is 𝜃, and the microstructure height is 𝐻0 for an SLDOE design working in the long-infrared waveband of 8∼12 μm and the angle range of 0◦ ∼30◦ using the ZnSe crystalline materials as the selected substrate. Finally, ABIADE can be determined when these values are certain. Therefore, as shown in Fig. 5, the horizontal coordinates (x, y) represent the incident angle and wavelength, respectively, and the z coordinate is the related surface microstructure height of the SLDOE. Fig. 5 shows that the corresponding microstructure heights differ under different pairs of incident angles and wavelengths, demonstrating that the micro-structure heights should be calculated under different conditions to satisfy different working situations. Therefore, the ABIADE for a given waveband and incident angle range can be calculated, enabling the selection of the optimal design wavelength and the microstructure height with respect to the design wavelength and incident angle to be calculated. Fig. 6 shows the influence on ABIADE in incident angles of 0◦ ∼30◦ in a waveband of 8∼12 μm. The ABIADE distribution can be obtained by the x coordinate of the selected wavelength, where the optimal design wavelength can be determined as ABIADE reaches its maximum. After obtaining the design wavelength, the microstructure height of the SLDOE can be calculated. Fig. 6 shows that the micro-structure height changes as the SLDOE changes under similar incident conditions. Additionally, the ABIADE varies with the incident angle and wavelength. After determining the SLDOE substrate material, the maximum value of ABIADE can also be determined. The SLDOE design wavelength can be obtained as 𝜆0 = 9.55 μm, and the corresponding maximum ABIADE value reaches 95.208%.
4.2.1. Analysis with different incident angles For different incident angles, the relation between the diffraction efficiency and wavelength can be calculated as shown in Fig. 8. In addition, the corresponding diffractive microstructure heights and minimum diffraction efficiencies through the whole waveband are shown in Table 2. In Fig. 8, the black, red, and green curves in each subgraph represent the diffraction efficiencies caused by the incident angles (10◦ , 20◦ and 30◦ ) over the whole waveband, where the diffraction efficiency is a function of the incident angle at wavelengths of 9.55, 10, and 10.13 μm, respectively. Fig. 8 shows that the SLDOE based on the ABIADE optimal method exhibits a better diffraction efficiency than that exhibited by the remaining two methods within the working waveband. Results of three designs (Row 1) can be presented in Table 2, including the design wavelength (Row 2), micro-structure height (Row 3) and the minimum diffraction efficiency under three incident angles situations (Row 4). It can be seen from Table 2 that considering the working angles based on three designs, the SLDOE maximized design based on ABIADE exhibits the highest diffraction efficiency at all the incident angles. However, the minimum diffraction efficiency considerably differs for different designs. For example, when the incident angle becomes 10◦ with an optimal design wavelength of 9.55 μm, the minimum diffraction efficiency is observed to be 86.510%, which is greater than those of the BIADE-based and mid-wavelength-based designs exhibiting values of 7.956% and 10.503%, respectively. In addition, the same trend can be observed for incident angles of 20◦ and 30◦ . Furthermore, the ABIADE-based design maximizes the SLDOE with a minimal surface microstructure height. 5
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Optics Communications 458 (2020) 124762
Table 1 Parameters for SLDOE under two incident situations with two designs. Working situation
Design method Height (μm) under normal incidence (0◦ ) Height (μm) under oblique incidence (45◦ ) BIADE (%) under normal incidence (0◦ ) BIADE (%) under oblique incidence (45◦ )
Working waveband
Visible 0.4∼0.7 μm
Optical materials
PMMA
Mid-wavelength based BIADE-based Mid-wavelength based BIADE-based Mid-wavelength based BIADE-based Mid-wavelength based BIADE-based Mid-wavelength based BIADE-based
0.55 0.564 1.114 1.116 0.904 0.905 90.201 90.147 90.286 90.254
Mid-infrared 3∼5 μm
Long-infrared 8∼12 μm
N-FK5
ZNSE
ZNSE
0.55 0.564 1.125 1.155 0.912 0.936 90.440 89.486 90.513 89.575
4.0 4.083 2.791 2.285 2.468 2.519 92.656 87.383 92.663 92.053
10.0 10.13 7.110 7.114 6.277 6.280 95.084 95.074 95.113 95.105
Table 2 Parameters based on three designs for three incident angles. Method
ABIADE-based Mid-wavelength-based BIADE-based
Wavelength/μm
9.55 10 10.13
Micro-structure height H/μm
6.777 6.978 7.207
𝜂min /% 10◦
20◦
30◦
86.510 78.554 76.007
83.610 74.972 72.255
78.219 68.567 65.609
Fig. 7. The diffraction efficiency versus incident angle and wavelength based on three methods: (a) angle bandwidth integrated average diffraction efficiency-; (b) mid-wavelength-; and (c) bandwidth integrated average diffraction efficiency-based designs.
efficiency is a function of wavelengths (9.55, 10, and 10.13 μm, respectively). Fig. 9 shows that the diffraction efficiency of the SLDOE designed using the ABIADE maximization method is considerably better than those of the remaining two designs in the working waveband.
4.2.2. Analysis with different wavelengths Fig. 9 shows the diffraction efficiency curves of the SLDOE in the working waveband and angle range for a waveband of 8∼12 μm. The microstructure height, minimum diffraction efficiency, and BIADE
Table 3 shows that the SLDOE design based on ABIADE has the highest BIADE of 94.678%, which is higher than the designs based on BIADE and mid-wavelength by 1.811% and 2.590%, respectively, at design wavelengths under an oblique incidence of 30◦ . Furthermore, the
calculation results are presented in Table 3. In Fig. 9, the black, red, and green curves in each subgraph represent the first-order SLDOE diffraction efficiencies, where the diffraction 6
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Optics Communications 458 (2020) 124762
Fig. 8. Relations between the diffraction efficiency and incident angles: (a) angle bandwidth integrated average diffraction efficiency-; (b) mid-wavelength-; and (c) bandwidth integrated average diffraction efficiency-based designs. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 9. Diffraction efficiencies of three single-layer diffraction optical element designs for (a) a normal incidence of 0◦ and (b) an oblique incidence of 30◦ . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Table 3 Parameters under three designs within waveband of 8∼12 μm. Method Wavelength /μm Micro-structure height H/μm Normal incidence
Oblique incidence @
30◦
ABIADE-based
Mid-wavelength based
BIADE-based
9.55
10.0
10.13
6.777
6.978
7.207
𝜂min / %
85.891
79.699
77.212
BIADE/ %
94.990
94.832
94.541
𝜂min /%
78.219
68.567
65.609
ABIADE/%
94.678
92.867
92.088
7
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Optics Communications 458 (2020) 124762
References
minimum diffraction efficiency based on this optimal design is higher than those of the remaining two designs with values of 9.652% and 12.61%, respectively. ABIADE can maximize the SLDOE design with a minimum micro-structure. In addition, in the working waveband, the SLDOE based on the ABIADE maximization design exhibits the maximum BIADE, which is higher than the diffraction efficiency for the BIADE-based and mid-wavelength-based designs with differences of 1.832% and 7.31%, respectively. This demonstrates that the ABIADEbased design is superior to the BIADE-based and mid-wavelength-based designs and that the diffraction efficiency of the SLDOE design based on the BIADE-based design is the worst.
[1] G.J. Swanson, Binary Optics Technology the Theory and Design of Multi-Level Diffractive Optical Elements, MIT Lincoln Laboratory technical report, 1989. [2] N. Davidson, A.A. Friesem, E. Hasman, Analytic design of hybrid diffractiverefractive achromats, Appl. Opt. 32 (25) (1993) 4770–4774. [3] B. Sabushinike, G. Horugavye, S. Habraken, Optimization of a multi-blaze grating in reflection using a free-form profile, Appl. Opt. 57 (18) (2018) 5048–5056. [4] B. Omri, G. Michael A., Multifunctional binary diffractive optical elements for structured light projectors, Opt. Express 26 (16) (2018) 21092–21107. [5] C. Wu, H. Gu, Z. Zhou, Q. Tan, Design of diffractive optical elements for sub-diffraction spot arrays with high light efficiency, Appl. Opt. 56 (3) (2017) 8816–8820. [6] T. Nakai, H. Ogawa, Research on Multi-Layer Diffractive Optical Elements and their Application To Camera Lenses, Optical Society of America, 2002. [7] C. Bigwood, A. Wood, Two-element lenses for military applications, Opt. Eng. 50 (12) (2011) 121705. [8] M.J. Riedl, Design example for the use of hybrid optical elements in the infrared, Appl. Opt. 35 (34) (1996) 6833–6834. [9] G.J. Swanson, W.B. Veldkamp, Diffractive optical elements for use in infrared systems, Opt. Eng. 28 (1989) 605–608. [10] P.P. Clark, C. Londono, Production of kinoforms by single point diamond machining, Opt. News. 15 (1989) 39–40. [11] A.P. Wood, P.J. Rogers, Diffractive optics in modern optical engineering, Proc. SPIE 5865 (2005) 83–97. [12] C. Xue, Q. Cui, Design of multilayer diffractive optical elements with polychromatic integral diffraction efficiency, Opt. Lett. 35 (7) (2010) 986–988. [13] H. Yang, C. Xue, C. Li, J. Wang, R. Zhang, Optimal design of multilayer diffractive optical elements with effective area method, Appl. Opt. 55 (25) (2018) 7126–7133. [14] M. Piao, Q. Cui, C. Zhao, B. Zhang, S. Mao, Y. Zhao, L. Zhao, Substrate material section method for multi-layer diffractive optics in a wide environment temperature range, Appl. Opt. 56 (10) (2017) 2826–2833. [15] B. Zhang, Q. Cui, M. Piao, Effect of substrate material selection on polychromatic integral diffraction efficiency for multi-layer diffractive optics in oblique situation, Opt. Commun. 415 (15) (2018) 156–163. [16] Y. Hu, Q. Cui, L. Zhao, M. Piao, PSF model for diffractive optical elements with improved imaging performance in dual-waveband infrared systems, Opt. Express 26 (21) (2018) 26845–26857. [17] D.C.O. Shea, T.J. Suleski, A.D. Kathman, D.W. Prather, Diffractive Optics Design Fabrication Test, SPIE, 2004. [18] D.A. Burali, G.M. Morris, Effects of diffraction efficiency on the modulation transfer function of diffractive lenses, Appl. Opt. 31 (22) (1992) 4389–4396. [19] X.D. Pei, Q.F. Cui, K.J. Leng, Effect of incident angle on diffraction efficiency of a two-layer diffractive optical element, Acta Optica Sinica. 29 (1) (2009) 120–125. [20] Y. Arieli, S. Ozeri, N. Eisenberg, S. Noach, Design of a diffractive optical element for wide spectral bandwidth, Opt. Lett. 23 (11) (1998) 823–824. [21] M. Bass, Optical Society of America, Handbook of Optics, II, McGraw-Hill Press, New York, 2010, Chapter 8.
5. Conclusion This study proposes an ABIADE concept and obtained the corresponding optimal wavelength by optimizing the ABIADE design within a given angle range and waveband and considering the influence of the incident angle range on the diffraction efficiency and microstructure height of the DOEs. Further, the corresponding surface micro-structure height can be obtained by selecting the optimal wavelength, allowing the realization of the optimal DOE design. The results denote that in the long-infrared waveband, the ABIADE-based designs for the SLDOE are better than the BIADE-based and mid-wavelength-based designs. The ABIADE-based optimization design is of considerable significance for obtaining the quantitative and optimal design of SLDOEs in hybrid imaging optical systems. Thus, the performance of different DOE-based imaging techniques can be compared. The presented method can be applied not only to conventional SLDOE, as done in this study, but also to the chromatically corrected HDOE and MLDOE designs. We hope that this study will serve as a guide for designing and evaluating the DOEs for practical applications. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Funding National Science Foundation of China (NSFC) (11634010, 61905195); China Postdoctoral Science Foundation (2018M643728); Natural Science Foundation of Shannxi Province, China (2019JQ-063); The Fundamental Research Funds for the Central Universities, China (310201911cx045).
8
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Diffractive optical element optimization under wide incident angle and waveband situations Shan Mao, Jianlin Zhao ∗ MOE Key Laboratory of Material Physics and Chemistry under Extraordinary Conditions, and Shaanxi Key Laboratory of Optical Information Technology, School of Science, Northwestern Polytechnical University, Xi’an 710072, China
ARTICLE
INFO
Keywords: Diffractive optical element Diffraction efficiency Incident angle range Optimal design
ABSTRACT The single-layer diffractive optical elements (SLDOEs) exhibit various advantages with respect to the design, configuration, fabrication, and cost. However, they also rapidly decrease the diffraction efficiency in a wide working waveband and incident angle range. In this study, we propose an optimal design and set up a mathematical model and derive expressions for SLDOE in actual applications to effectively solve this problem. Initially, an optimal SLDOE design can be obtained by establishing the mathematical model and diffraction efficiency expressions for SLDOEs with respect to the incident angle. Finally, the diffraction efficiency within an incident angle range is presented based on three typical designs, and their influences are analyzed. The results denote that the angle bandwidth integrated average diffraction efficiency (ABIAED)-based design can improve the diffraction efficiency over the entire waveband and range of incident angles, while improving the ease of processing of SLDOEs with a minimum microstructure height.
1. Introduction Diffractive optical elements (DOEs) exhibit special dispersion and thermal properties that considerably enhance their usefulness in several applications [1,2]. When size or weight of the optical system is crucial, DOEs provide substantial advantages and can be used to design an aberration-minimizing compact imaging system thinner and lighter than a system manufactured using traditional reflective or refractive optical elements. DOEs can provide additional design freedom for various electro-optical instrument applications, such as while being used in conjunction with other optical components or producing desired effects on their own depending on what the optical designers are attempting to achieve for special design concepts [3–5]. The commonly used hybrid optical lens comprises a conventional refractive or reflective element as the substrate combined with a DOE micro-structure on one surface. In this system, DOEs can be used to correct image aberrations, especially color aberrations, considerably similar to aspheric surfaces or additional refractive lenses [6]. Thus, they can be extensively applied to visible and infrared cameras for aberration correction in militaries, high-end optical systems, and other such applications [7–9]. Current research progresses have made it possible to design and manufacture DOEs with high micro-structure precision [10] and diffraction efficiency. Additionally, many studies related to the DOE implementation methods and performance evaluation have been published [11–15]. Currently, the effective energy imaged using DOE can be attributed to a first-order diffractive beam. For the remaining orders, ∗
the beam is diffused throughout the image plane, resulting in the creation of harmful energy. To address this problem, the detector uses a point spread function (PSF) model for image collection, and high diffractive order energies converge to the image plane. The DOE image quality will be substantially improved when the effective energy of the system is combined with the design method and the PSF image processing model [16]. Typical types of DOE include single-layer DOEs (SLDOE), multilayer DOEs (MLDOE), and harmonic DOEs (HDOE). SLDOE is the easiest to design and manufacture; however, SLDOEs exhibit a significant limitation with respect to the strong diffraction efficiency sensitivity caused by the wavelength derived from the design wavelength and working incident angle. Diffraction efficiency is a major concern for optical engineers; therefore, effective designs should be established to improve the diffraction efficiency. Recent research has resulted in some excellent SLDOE designs based on light being incident perpendicularly or under a certain incident angle to the SLDOE surface. Obviously, all these designs do not consider the influence of the incident angle ranges on the phase delay and diffraction efficiency. Two important designs are obtained based on the mid-wavelength and bandwidth integrated average diffraction efficiency (BIADE) solutions [17]. The SLDOEs applied in hybrid optical systems are always under oblique incidence conditions on its surface, indicating that the design optimization and diffraction efficiency analysis of SLDOEs are universally significant and practical under oblique
Corresponding author. E-mail address: [email protected] (S. Mao).
https://doi.org/10.1016/j.optcom.2019.124762 Received 6 August 2019; Received in revised form 8 October 2019; Accepted 13 October 2019 Available online 17 October 2019 0030-4018/© 2019 Elsevier B.V. All rights reserved.
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Optics Communications 458 (2020) 124762
Fig. 1. The incident angle distributions on a single-layer diffractive optical element surface for (a) a large aperture system and (b) a large field-of-view system.
incidence conditions. This is the case even for a large view angle or aperture angle, which is practical, especially in zoom hybrid optical systems typically applied to high-end optical systems. Therefore, the major objective of this study is to present an optimal SLDOE design and the related diffraction efficiency optimization using design wavelength selection and DOE parameter calculations. In this study, we propose a concept for angle bandwidth integrated average diffraction efficiency (ABIADE) and establish a corresponding mathematical model. Subsequently, we performed SLDOE optimization based on the aforementioned concept and model. The optimized design can ensure high diffraction properties for a SLDOE over a wide range of incident angle, ensuring the imaging quality of a hybrid optical system. Furthermore, we explored a novel method for evaluating SLDOEs in a broad waveband range with large incident angles. This method can expand the working waveband range when compared with that in the traditional designs. In addition, optical engineers can apply the analysis described in this study to both the MLDOE and HDOE lenses.
the SLDOE diffraction efficiency based on the mid-wavelength design at several incident wavelengths. When the long-infrared waveband is selected as the working waveband and 0◦ ∼45◦ is selected as the incident angle range, the diffraction efficiency can be evaluated; the results are presented in Fig. 2. Fig. 2(a) shows the relation among the incident angle, wavelength, and corresponding diffraction efficiency, Fig. 2(b) shows the relation between the incident angle and overall BIADE with respect to the long-infrared band, and Fig. 2(c) shows the relation between the wavelength and diffraction efficiency under different incident situations. As can be observed, Fig. 2(b) and (c) are derived from Fig. 2(a). Fig. 2 denotes that when the incident angle gradually increases, the diffraction efficiency initially decreases slowly, indicating that the SLDOE can be used in this angle range. However, the diffraction efficiency drastically declines as the incident angle increases further. Therefore, the incident angle formed by the incident light and SLDOE surface decreases the diffraction efficiency to the decreased diffraction efficiency obtained by design wavelength deviation. Hence, the decrease in the incident-angle-driven efficiency is an important problem that cannot be ignored in current SLDOE designs.
2. Problem statement Diffraction efficiency is an important parameter that determines the working waveband and DOE application. The modulation transform function (MTF) is directly affected by the BIADE for a hybrid optical system or lens [18]. The diffraction efficiency drastically decreases when the conventional DOE’s working waveband becomes wider and the incident angle increases, reducing the overall BIADE waveband and the MTF of the hybrid optical system. Therefore, the high diffraction efficiency requirements can be satisfied by reselecting the design wavelengths for achieving microstructure optimization. In hybrid imaging optical systems, a certain angle of light is incident on the SLDOE surface when the light is incident on the SLDOE itself and when the SLDOE works within a certain incident angle range in a hybrid optical system. Fig. 1 illustrates a hybrid imaging lens using a SLDOE, with Fig. 1(a) denoting the incident angle of a large aperture and Fig. 1(b) denoting the incident angle of a large field of view. As depicted in Fig. 1, the normal incident angle of a beam on the SLDOE surface is one of the simplest parameters. Generally, the beam transmission and diffraction efficiency at a certain incident angle should be considered. Therefore, an optical system is always with a large angle of view or a large aperture angle; then, the diffraction efficiency and the BIADE analysis of the SLDOE in an oblique incident angle range exhibits universal significance and is considerably practical. Fig. 2 shows the influence curves of the incident angle on
3. Optimization design The analyses presented in Section 2 demonstrate that the incident angle must be considered in SLDOE optimization to satisfy the high diffraction efficiency requirements within an incident angle range to ensure high imaging quality for hybrid imaging optical systems. A set of values should be obtained with respect to all the working wavelength and incident angle pair values for an SLDOE, i.e., (𝜆1, 𝜃1 ), (𝜆2, 𝜃2 ), (𝜆3, 𝜃3 ), … , (𝜆𝑖, 𝜃𝜄 ), to achieve high diffraction efficiency. Thus, by considering different working wavelength and incident angle pairs during the calculation, maximum diffraction efficiency can be obtained for a certain SLDOE. Subsequently, the microstructure height can be calculated based on the unique optimal design wavelength to achieve maximum BIADE [1]. SLDOEs achieve an optimal design using an ABIADE-based design at a certain incident angle and waveband. Further, the optimal design wavelength and diffractive microstructure parameters can be obtained based on the working waveband and incident angle range of a hybrid optical system, optimizing the maximum ABIADE value. In this section, SLDOE is optimized to achieve maximum ABIADE, where the optimal wavelength and microstructure height are considered to be the intermediate values. The process of maximizing ABIADE can be divided into the 2
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Optics Communications 458 (2020) 124762
Fig. 2. Wavelength and incident angle diffraction efficiencies. (a) Diffraction efficiency that can be attributed to the wavelength and incident angle; (b) angle bandwidth integrated average diffraction efficiency that can be attributed to the incident angle; and (c) the diffraction efficiency for different wavelengths at different incident angles.
following three parts: determining the maximum ABIADE; optimizing the design wavelength; and calculating the micro-structure height. Because the incident angle range has been presented at the beginning of the optical system design, we do not optimize the incident angle in this section. Fig. 3 depicts the SLDOE optimization design process. In Fig. 3, 𝜂3 represents the ABIADE maximum value of 𝜂 𝑚 (𝜆, 𝜃), where 𝜆 denotes the design wavelength and 𝜃 denotes the incident angle. 𝜂1 and 𝜂2 denote the diffraction efficiencies over all the incident angles and waveband. Further, 𝜆max represents the maximum and minimum wavelengths, and 𝜃max and 𝜃min represent the maximum and minimum incident angles, respectively. The specific optimization design idea is that the ABIADE can be calculated point–point with the angle range waveband considering the incident angle range and waveband; additionally, the ABIADE values corresponding to different incident angles and wavelengths is compared to obtain a one-to-one correspondence for the maximum ABIADE. Finally, the design wavelength can be determined. According to this design wavelength, the diffraction microstructure height of the SLDOE can be calculated; thus, the optimal SLDOE design can be achieved. Based on Fourier optics and scalar approximation theory, we present the optimization analysis of SLDOE. When we do not consider material absorption, reflection, and scattering of incident light, the diffraction efficiency of a continuous surface can be expressed as [1] [ ]2 𝜂𝑚 = sinc(𝑚 − 𝜙(𝜆, 𝜃)) , (1)
Fig. 3. The optimal flow for a single-layer diffractive optical element.
where sinc(𝑥) = sin(𝜋𝑥)∕(𝜋𝑥), 𝜙(𝜆, 𝜃) is the phase delay, 𝜃 is the incident angle, and 𝜆 represents the wavelength. m denotes the diffraction order, and only the first diffraction order of 𝑚 = 1 is usually considered to denote the effective energy. When determining the SLDOE material, incident angle 𝜃, and wavelength 𝜆, the phase delay should be equal to the diffraction order to
ensure 100% diffraction efficiency. Therefore, the corresponding phase delay from Eq. (1) can be expressed as [19] [√ ] 𝐻 𝜙(𝜆, 𝜃) = 𝑛2 (𝜆) − 𝑛2𝑚 (𝜆) sin2 𝜃 − 𝑛𝑚 (𝜆) cos 𝜃 , (2) 𝜆 3
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where H is the micro-structure height, n(𝜆) is the refractive index of the substrate material, and 𝑛𝑚 (𝜆) is the refractive index of the medium. Thus, by substituting Eq. (2) into Eq. (1), the diffraction efficiency of SLDOE can be expressed as { [√ ]} 𝐻 𝑛2 (𝜆) − 𝑛2𝑚 (𝜆) sin2 𝜃 − 𝑛𝑚 (𝜆) cos 𝜃 . (3) 𝜂𝑚 (𝜆, 𝜃) = sinc2 1 − 𝜆 To maximize the first-order diffraction efficiency of a hybrid tical system, the phase delay because of its phase height should come an integral multiple of the design wavelength. Subsequently, microstructure height can be obtained as [√ ] 𝐻 𝑛2 (𝜆0 ) − 𝑛2𝑚 (𝜆0 ) sin2 𝜃 − 𝑛𝑚 (𝜆0 ) cos 𝜃 = 𝜆0 .
opbethe
(4)
When the design wavelength 𝜆0 exhibits different values in the working waveband and when the incident angle 𝜃 is any value within the angle range, the micro-structure height H will lead to a series result. The important indicators for designing and evaluating the SLDOE should consider factors, such as the working waveband, because the diffraction efficiency at a single wavelength cannot denote the overall performance. Additionally, the incident angle of light on the microstructure surface has been included in the original design specifications. Therefore, an ABIADE concept has been proposed in this study. The incident light in an optical system is generally not from a monochromatic source; therefore, it is meaningful to evaluate the BIADE that affects system imaging. In case of SLDOE, ABIADE refers to the overall imaging performance with respect to the incident angle range and working waveband, which can be expressed as 𝜂 𝑚 (𝜆, 𝜃) = =
𝜃
𝜆
𝜃
𝜆
max max 1 𝜂𝑚 𝑑𝜆𝑑𝜃 ∫ ∫ 𝜆max − 𝜆min 𝜃min 𝜆min
,
max max 1 sinc2 [1 − 𝜙(𝜆, 𝜃)] 𝑑𝜆𝑑𝜃 𝜆max − 𝜆min ∫𝜃min ∫𝜆min
Fig. 4. Relations between the refractive index and wavelength in different wavebands for several optical materials.
based on the BIADE requirements, ensuring the MTF of the system and yielding optimal image quality for the hybrid optical system under normal incidence. However, when the beam is incident onto the SLDOE at a certain angle, ABIADE is the corresponding value of the available waveband within a large incident angle range. Finally, the diffraction efficiency corresponding to different designs varies, and the differences in diffraction efficiency can be expressed as
(5)
⎧𝛥𝑠 = 𝜂 (𝜃1 , 𝜃2 , … 𝜃𝑖 ; 𝜆1 , 𝜆2 , … 𝜆𝑖 ) 𝐴𝐵𝐼𝐴𝐷𝐸 ⎪ 1 − 𝜂 𝑚𝑖𝑑−𝑤𝑎𝑣𝑒𝑙𝑒𝑛𝑔𝑡ℎ (𝜃1 , 𝜃2 , … 𝜃𝑖 ; 𝜆1 , 𝜆2 , … 𝜆𝑖 ) ⎪ . ⎨ ⎪𝛥𝑠 2 = 𝜂 𝐴𝐵𝐼𝐴𝐷𝐸 (𝜃1 , 𝜃2 , … 𝜃𝑖 ; 𝜆1 , 𝜆2 , … 𝜆𝑖 ) ⎪ − 𝜂 𝐵𝐼𝐴𝐷𝐸 (𝜃1 , 𝜃2 , … 𝜃𝑖 ; 𝜆1 , 𝜆2 , … 𝜆𝑖 ) ⎩
where 𝜆max and 𝜆min represent the maximum and minimum wavelengths, respectively, and 𝜃max and 𝜃min represent the maximum and minimum incident angles, respectively. In case of a conventional SLDOE, BIADE is observed to directly affect the imaging quality of the hybrid optical system. An optical imaging system’s modulation transfer function (𝑀𝑇 𝐹 𝑝𝑜𝑙𝑦 ) is the product of BIADE and can be expressed as [20] { 𝜂 ⋅ 𝑀𝑇 𝐹𝑖𝑑𝑒𝑎𝑙 (𝑓𝑥 , 𝑓𝑦 ), 𝑓𝑥 ≠ 0, 𝑓𝑦 ≠ 0 . (6) 𝑀𝑇 𝐹𝑝𝑜𝑙𝑦 = 1, 𝑓𝑥 = 𝑓𝑦 = 0
Generally, the design process of a hybrid optical system can be divided into the following two steps: the first step is the optimization design and calculations for the SLDOE and the second step is the optimization design for the hybrid optical system with respect to the SLDOE. 4. Simulation and analysis
In Eq. (6), 𝑀𝑇 𝐹 𝑖𝑑𝑒𝑎𝑙 denotes the ideal MTF of the hybrid optical system based on the optical design software and 𝑓𝑥 and 𝑓𝑦 denote the frequencies toward the x and y directions, respectively. Therefore, by considering the incident angle and substituting Eq. (5) into Eq. (6), the real MTF for a hybrid optical system can be expressed as { 𝜂 𝑚 (𝜆, 𝜃) ⋅ 𝑀𝑇 𝐹𝑖𝑑𝑒𝑎𝑙 (𝑓𝑥 , 𝑓𝑦 ), 𝑓𝑥 ≠ 0, 𝑓𝑦 ≠ 0 . (7) 𝑀𝑇 𝐹𝑝𝑜𝑙𝑦 = 1, 𝑓𝑥 = 𝑓𝑦 = 0
Optical materials, such as optical plastics and glasses in the visible waveband and crystalline materials in the infrared waveband, exhibit significantly different characteristics, resulting in large diffraction efficiency and micro-structure differences for SLDOEs. The optical handbook [21] states that the refractive indices and Abbe number of the infrared crystal materials are significantly larger than those of the optical plastics and glasses. Additionally, using infrared wavebands as an example, the optical performance for the same crystal material differs while working in the long-infrared (8∼12 μm) and mid-infrared wavebands (3∼5 μm). According to the optical software and material dispersion expressions, Fig. 4 simulates the relations between the refractive index and wavelength in different wavebands for several optical materials. Although optical plastics are not commonly used in highperformance optical lenses for hybrid optical systems, DOEs with high-precision continuous surfaces can be produced based on the development of the single-point diamond turning (SPDT) technique. In addition, the infrared crystal material has a larger Abbe number than those of glasses and optical plastics. When a diffractive–refractive hybrid lens satisfies achromatic and thermal conditions under infrared light, DOE exhibits a small distribution of optical power. Additionally, the number of diffraction cycle rings is significantly lower than that
When the light is perpendicularly incident to the SLDOE, based on Eq. (5), BIADE can be approximated from Eq. (5) as 𝜂 𝑚=1 ≈ 1 +
𝜋2 𝜋2 2 (𝜆min + 𝜆max − 𝜆0 ) − (𝜆 + 𝜆min 𝜆max + 𝜆2max ). 3𝜆0 9𝜆20 min
(8)
When the design wavelength 𝜆0 is determined, the BIADE of SLDOE can also be determined; thus, the selection of design wavelength can directly affect the optical system’s MTF. On both sides of Eq. (8), the design wavelength 𝜆0 can be calculated based on as 𝜆0 =
2(𝜆2min + 𝜆max 𝜆min + 𝜆2max ) 3(𝜆max + 𝜆min )
.
(10)
(9)
When the design wavelength can satisfy Eq. (9), BIADE reaches its maximum and minimally affects the image quality. Further, the available working waveband of the optical system can be determined 4
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Fig. 6. Relation between the angle bandwidth integrated average diffraction efficiency and wavelength within the whole incident angle range.
Fig. 5. Micro-structure height caused by the incident wavelength and incident angle.
4.2. Analysis
in DOEs based on optical glasses and plastics. Therefore, we select a crystalline material that can work in a long-infrared waveband for performing simulation and analysis. Based on the material characteristics and Eq. (9), the design wavelength based on two traditional methods and microstructure heights under normal and oblique incident angles can be calculated, and the results are presented in Table 1. As presented in Table 1, the micro-structures differ under two incident situations. The microstructure heights and their differences in case of the long-infrared waveband are much greater than those for the visible and middle-infrared SLDOE.
The comprehensive effects of incident angle and wavelength on the diffraction efficiency was analyzed for different design wavelengths (based on different methods shown in Fig. 7), as shown in Fig. 7. Fig. 7 shows the diffraction efficiency effects caused by the incident angle and wavelength for ABIADE-, mid-wavelength-, and BIADE-based designs with design wavelengths of 9.55, 10, and 10.13 μm, respectively. The comparisons in Fig. 7 show that the overall diffraction efficiency of the ABIADE design is higher than those of the BIADE or midwavelength designs and that the performance has been significantly improved. As presented in Table 1, in the working angle range of 0◦ ∼30◦ and a waveband of 8∼12 μm, the diffraction efficiency can reach its maximum with the ABIADE-based design, greater than those of the BIADE- and mid-wavelength-based designs.
4.1. Optimal design With respect to the pairs of wavelengths and incident angles, the different microstructure heights can be calculated based on Eqs. (3) and (4); thus, the diffraction efficiency of the SLDOE exhibits minimal influence on the MTF and image quality for a hybrid optical system. Based on the optimal design theory, we assume that the design wavelength is 𝜆0 , the incident angle is 𝜃, and the microstructure height is 𝐻0 for an SLDOE design working in the long-infrared waveband of 8∼12 μm and the angle range of 0◦ ∼30◦ using the ZnSe crystalline materials as the selected substrate. Finally, ABIADE can be determined when these values are certain. Therefore, as shown in Fig. 5, the horizontal coordinates (x, y) represent the incident angle and wavelength, respectively, and the z coordinate is the related surface microstructure height of the SLDOE. Fig. 5 shows that the corresponding microstructure heights differ under different pairs of incident angles and wavelengths, demonstrating that the micro-structure heights should be calculated under different conditions to satisfy different working situations. Therefore, the ABIADE for a given waveband and incident angle range can be calculated, enabling the selection of the optimal design wavelength and the microstructure height with respect to the design wavelength and incident angle to be calculated. Fig. 6 shows the influence on ABIADE in incident angles of 0◦ ∼30◦ in a waveband of 8∼12 μm. The ABIADE distribution can be obtained by the x coordinate of the selected wavelength, where the optimal design wavelength can be determined as ABIADE reaches its maximum. After obtaining the design wavelength, the microstructure height of the SLDOE can be calculated. Fig. 6 shows that the micro-structure height changes as the SLDOE changes under similar incident conditions. Additionally, the ABIADE varies with the incident angle and wavelength. After determining the SLDOE substrate material, the maximum value of ABIADE can also be determined. The SLDOE design wavelength can be obtained as 𝜆0 = 9.55 μm, and the corresponding maximum ABIADE value reaches 95.208%.
4.2.1. Analysis with different incident angles For different incident angles, the relation between the diffraction efficiency and wavelength can be calculated as shown in Fig. 8. In addition, the corresponding diffractive microstructure heights and minimum diffraction efficiencies through the whole waveband are shown in Table 2. In Fig. 8, the black, red, and green curves in each subgraph represent the diffraction efficiencies caused by the incident angles (10◦ , 20◦ and 30◦ ) over the whole waveband, where the diffraction efficiency is a function of the incident angle at wavelengths of 9.55, 10, and 10.13 μm, respectively. Fig. 8 shows that the SLDOE based on the ABIADE optimal method exhibits a better diffraction efficiency than that exhibited by the remaining two methods within the working waveband. Results of three designs (Row 1) can be presented in Table 2, including the design wavelength (Row 2), micro-structure height (Row 3) and the minimum diffraction efficiency under three incident angles situations (Row 4). It can be seen from Table 2 that considering the working angles based on three designs, the SLDOE maximized design based on ABIADE exhibits the highest diffraction efficiency at all the incident angles. However, the minimum diffraction efficiency considerably differs for different designs. For example, when the incident angle becomes 10◦ with an optimal design wavelength of 9.55 μm, the minimum diffraction efficiency is observed to be 86.510%, which is greater than those of the BIADE-based and mid-wavelength-based designs exhibiting values of 7.956% and 10.503%, respectively. In addition, the same trend can be observed for incident angles of 20◦ and 30◦ . Furthermore, the ABIADE-based design maximizes the SLDOE with a minimal surface microstructure height. 5
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Table 1 Parameters for SLDOE under two incident situations with two designs. Working situation
Design method Height (μm) under normal incidence (0◦ ) Height (μm) under oblique incidence (45◦ ) BIADE (%) under normal incidence (0◦ ) BIADE (%) under oblique incidence (45◦ )
Working waveband
Visible 0.4∼0.7 μm
Optical materials
PMMA
Mid-wavelength based BIADE-based Mid-wavelength based BIADE-based Mid-wavelength based BIADE-based Mid-wavelength based BIADE-based Mid-wavelength based BIADE-based
0.55 0.564 1.114 1.116 0.904 0.905 90.201 90.147 90.286 90.254
Mid-infrared 3∼5 μm
Long-infrared 8∼12 μm
N-FK5
ZNSE
ZNSE
0.55 0.564 1.125 1.155 0.912 0.936 90.440 89.486 90.513 89.575
4.0 4.083 2.791 2.285 2.468 2.519 92.656 87.383 92.663 92.053
10.0 10.13 7.110 7.114 6.277 6.280 95.084 95.074 95.113 95.105
Table 2 Parameters based on three designs for three incident angles. Method
ABIADE-based Mid-wavelength-based BIADE-based
Wavelength/μm
9.55 10 10.13
Micro-structure height H/μm
6.777 6.978 7.207
𝜂min /% 10◦
20◦
30◦
86.510 78.554 76.007
83.610 74.972 72.255
78.219 68.567 65.609
Fig. 7. The diffraction efficiency versus incident angle and wavelength based on three methods: (a) angle bandwidth integrated average diffraction efficiency-; (b) mid-wavelength-; and (c) bandwidth integrated average diffraction efficiency-based designs.
efficiency is a function of wavelengths (9.55, 10, and 10.13 μm, respectively). Fig. 9 shows that the diffraction efficiency of the SLDOE designed using the ABIADE maximization method is considerably better than those of the remaining two designs in the working waveband.
4.2.2. Analysis with different wavelengths Fig. 9 shows the diffraction efficiency curves of the SLDOE in the working waveband and angle range for a waveband of 8∼12 μm. The microstructure height, minimum diffraction efficiency, and BIADE
Table 3 shows that the SLDOE design based on ABIADE has the highest BIADE of 94.678%, which is higher than the designs based on BIADE and mid-wavelength by 1.811% and 2.590%, respectively, at design wavelengths under an oblique incidence of 30◦ . Furthermore, the
calculation results are presented in Table 3. In Fig. 9, the black, red, and green curves in each subgraph represent the first-order SLDOE diffraction efficiencies, where the diffraction 6
S. Mao and J. Zhao
Optics Communications 458 (2020) 124762
Fig. 8. Relations between the diffraction efficiency and incident angles: (a) angle bandwidth integrated average diffraction efficiency-; (b) mid-wavelength-; and (c) bandwidth integrated average diffraction efficiency-based designs. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 9. Diffraction efficiencies of three single-layer diffraction optical element designs for (a) a normal incidence of 0◦ and (b) an oblique incidence of 30◦ . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Table 3 Parameters under three designs within waveband of 8∼12 μm. Method Wavelength /μm Micro-structure height H/μm Normal incidence
Oblique incidence @
30◦
ABIADE-based
Mid-wavelength based
BIADE-based
9.55
10.0
10.13
6.777
6.978
7.207
𝜂min / %
85.891
79.699
77.212
BIADE/ %
94.990
94.832
94.541
𝜂min /%
78.219
68.567
65.609
ABIADE/%
94.678
92.867
92.088
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S. Mao and J. Zhao
Optics Communications 458 (2020) 124762
References
minimum diffraction efficiency based on this optimal design is higher than those of the remaining two designs with values of 9.652% and 12.61%, respectively. ABIADE can maximize the SLDOE design with a minimum micro-structure. In addition, in the working waveband, the SLDOE based on the ABIADE maximization design exhibits the maximum BIADE, which is higher than the diffraction efficiency for the BIADE-based and mid-wavelength-based designs with differences of 1.832% and 7.31%, respectively. This demonstrates that the ABIADEbased design is superior to the BIADE-based and mid-wavelength-based designs and that the diffraction efficiency of the SLDOE design based on the BIADE-based design is the worst.
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5. Conclusion This study proposes an ABIADE concept and obtained the corresponding optimal wavelength by optimizing the ABIADE design within a given angle range and waveband and considering the influence of the incident angle range on the diffraction efficiency and microstructure height of the DOEs. Further, the corresponding surface micro-structure height can be obtained by selecting the optimal wavelength, allowing the realization of the optimal DOE design. The results denote that in the long-infrared waveband, the ABIADE-based designs for the SLDOE are better than the BIADE-based and mid-wavelength-based designs. The ABIADE-based optimization design is of considerable significance for obtaining the quantitative and optimal design of SLDOEs in hybrid imaging optical systems. Thus, the performance of different DOE-based imaging techniques can be compared. The presented method can be applied not only to conventional SLDOE, as done in this study, but also to the chromatically corrected HDOE and MLDOE designs. We hope that this study will serve as a guide for designing and evaluating the DOEs for practical applications. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Funding National Science Foundation of China (NSFC) (11634010, 61905195); China Postdoctoral Science Foundation (2018M643728); Natural Science Foundation of Shannxi Province, China (2019JQ-063); The Fundamental Research Funds for the Central Universities, China (310201911cx045).
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