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Abstract
This paper proposes a method that combines the effective area method and the oblique factors to analyze and design dual-layer diffractive optical elements at large incident angles. The method considers the effects of shadow and shield on the diffraction efficiency, as well as the correction of phase delay due to oblique incidence. The relationships among the diffraction efficiency, incident wavelengths, incident angles and period widths were analyzed with our method. A detailed comparison of the proposed method with the scalar diffraction theory is presented. The method was validated by numerical simulations of vector diffraction theory and showed similar diffraction efficiency distributions at different wavelengths and incident angles. The method provides a simple and efficient way to design and apply DLDOEs.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Diffractive optical elements (DOEs) are optical components that modulate the phase and/or amplitude of an incident light wave through a periodic or aperiodic microstructure. DOEs have many advantages over conventional refractive or reflective optics, such as low weight, compact size, high diffraction efficiency, and multifunctionality [1–4]. DOEs can be used for various applications, such as beam shaping [5,6], beam splitting [7,8], wavelength multiplexing [9], and imaging [10,11].
Perhaps ever since the initial application of gratings, the aspiration to compel the whole incident light to diffract into a specific order coincided with the aspiration to preserve flawless wavefront.
DLDOEs are a special type of DOEs that consists of two layers of DOEs with different dispersion materials [12]. DLDOEs can achieve high diffraction efficiency and achromatic performance in a wide spectral range and a wide field of view. DLDOEs have been widely used for dual-waveband or multi-waveband infrared imaging systems, where chromatic aberration and diffraction efficiency are critical factors for image quality [13].
Currently, scalar diffraction theory (SDT) is commonly employed for the analysis and design of DLDOEs when the period width is large enough [14,15]. However, the SDT has limitations when it comes to calculating the phase delay of a DLDOEs at large incident angles [16]. The scalar diffraction theory assumes that the incident angle is small, typically within the range of plus or minus five degrees. When the incident angle exceeds this range, the scalar diffraction theory fails to accurately capture the complex wavefront changes and diffraction effects.
To overcome this limitation, one possible approach is to use rigorous diffraction theories, such as the rigorous coupled-wave analysis (RCWA) or the finite difference time domain (FDTD) method. These methods consider the full vector nature of light and can accurately calculate the diffraction effects at larger incident angles. The rigorous vector method, although accurate, can be computationally expensive, especially for complex diffractive optical elements with large periodic widths. In such cases, the use of scalar diffraction theory with some corrections and modifications can be a more efficient approach.
We previously investigated the impact of shield and shadow effects on the diffraction efficiency of dual-layer diffraction optical elements, and proposed an equivalent area method (EAM) to correct the computation results of scalar diffraction theory [17]. You can accurately calculate the diffraction efficiency of the multi-layer diffractive optical elements with EAM, including those with periodic widths of tens to hundreds of micrometers. This method yielded satisfactory results under normal incidence conditions, but deviated significantly from reality as the incident angle further increased.
In this paper, we present a simple method to calculate the diffraction efficiency of DLDOEs at large incident angles. We incorporate the oblique factor into the calculation of the diffraction efficiency of the double-layer diffractive optical elements, further improving the diffraction efficiency analysis model of the double-layer diffractive optical elements. This method uses three effects to correct the analysis results of the scalar diffraction theory, and can obtain analysis results close to the vector diffraction theory.
The rest of this paper is organized as follows: Section 2 describes our proposed method and its derivation. We discuss the limitations of the scalar diffraction theory at large incident angles and highlight the need for more advanced diffraction models. Section 3 presents some numerical examples and comparisons with other methods. We discuss the advantages and limitations of our method. Section 4 concludes this paper and suggests some future work.
2. Methods
DLDOEs consist of two harmonic diffractive elements, a schematic diagram of DLDOEs is shown in Fig. 1(a), where blue represents the first layer of harmonic diffractive optical elements, and green represents the second layer of harmonic diffractive optical elements. By merging two diffractive elements with distinct phase profiles, DLDOEs can achieve intricate wavefront modulation and produce desired optical functions.
Fig. 1. Determination of shadow and shield lengths in the EAM method based on diffraction angle and light propagation. (a) Structure of the diffraction element. (b) Light propagation diagram for the first layer diffraction angle greater than the incident angle. (c) Light propagation diagram for the first layer diffraction angle less than the incident angle
The diffraction order distribution of the DLDOEs is the same as that of the single-layer diffractive optical element, and both are determined by the grating equation. However, the diffraction efficiency distribution of each diffraction order of the DLDOEs is very different from that of the single-layer diffractive optical element. The DLDOEs use two layers of larger microstructure heights to obtain a high diffraction efficiency over a wide wavelength band.
The EAM is used to recalculate the diffraction efficiency of DLDOEs, considering both the shield effect and the shadow effect [17]. This method allows for a more accurate evaluation of the diffraction efficiency, particularly when dealing with elements with finite period lengths. The shield effect refers to the reduction in diffraction caused by the presence of other elements or structures close to the diffractive element, whereas the shadow effect accounts for the optical shadow cast by these elements. When the shield and shadow effects are considered, EAM provides a more comprehensive analysis of the diffraction efficiency of DLDOEs.
To fully account for the shield and shadow in DLDOEs, two conditions based on diffraction angles can be considered. The first case is when the diffraction angle is larger than the incident angle, as shown in Fig. 1(b); The second case is when the diffraction angle is smaller than the incident angle, as shown in Fig. 1(c).
Figure 1(b) and (c) are magnified views of the dashed box in Fig. 1(a), for generality, the curvature of the first-layer and second-layer HDEs interface is ignored and treated as a plane. The microstructure period width of DLDOE is T, the microstructure height of the first layer element is H1, and the microstructure height of the second layer element is H2. t1, t2, t3, and t4 are the shadow lengths of the four corners, respectively, and h1 and h2 are the effective microstructure heights after being shielded. ${\theta _0}$ and ${\theta _1}$ are the incident angle and diffractive angle corresponding to the maximum diffraction efficiency.
The following is the sign convention for the incident angle and the diffraction angle: when the light ray lies on the left of the normal to the diffractive element, the sign is negative. When the light ray lies on the right of the normal to the diffractive element, the sign is positive. In Fig. 1(b), the incident angle has a negative value, and the diffraction angle has a positive value. In Fig. 1(c), the incident angle has a positive value, and the diffraction angle has a negative value.
2.1 Shadow effect when the first diffraction angle is larger than the incident angle
The diffraction condition when the first diffraction angle is larger than incident angle is shown in Fig. 1(b). On the right side of the same period, the light is chiefly obstructed by the second layer diffractive optical element and casts a shadow, causing t1 to enlarge. On the left side of the same period, the light is chiefly constrained by the bottom left corner of the first layer diffractive element and casts a shadow, causing t4 to enlarge. We assume that the light rays come from the points that are not shaded, and the shadow length of each part is given by the coordinates of the points where the rays intersect the diffractive element. The shadow lengths can be derived from geometric relations:
2.2 Shadow effect when the first diffraction angle is smaller than the incident angle
As the incident angle increases further, the diffracted light shifts to the left. At this time, the light on the right side of the same period is chiefly obstructed by the first layer element and casts a shadow, causing t1 and t3 to enlarge. Before the diffracted light moves to the normal, that is, when diffraction angle is greater than zero, the light on the left side of the same period is mainly blocked by the second layer diffractive element and produces a shadow, which is still the same as the situation on the left side of the same period in Fig. 1(b).
As the diffraction angle diminishes further, and passes the normal of the diffraction element to acquire a negative value, the case is illustrated on the left of Fig. 1(c). In this instance, t1 and t3 are still calculated from the previous Formula (3), and t2 and t4 are derived from the subsequent formula.
The diffraction efficiency of an optical system can be described by the sinc2 function, which is given by:
where: sinc(x) is the cardinal sin function, given by sinc(x) = sin(x)/x. m is the diffraction order and $\phi$ is the phase delay.The four shadow lengths affect the diffraction efficiency in the form of sinc^2(t/T)
The sinc2 function illustrates that the central peak of the diffraction pattern has the highest diffraction efficiency, which gradually decreases toward the edges. In the context of shadow lengths, this equation shows that the magnitude of the diffraction efficiency depends on the ratio between the shadow length t and the microstructure period length scale T. The longer the shadow length compared to T, the more the diffraction pattern spreads out. Conversely, as the shadow length decreases compared to T, the diffraction efficiency increases.
2.3 Shield effect of DLDOEs
In diffraction elements, such as gratings or other diffractive optical structures, the microstructure height plays a crucial role in determining their diffraction properties. However, when obstructions are present that partially shield or obstruct certain regions of the microstructures, it decreases their effective height.
This shield effect can significantly affect the diffraction efficiency and intensity distribution of light passing through the diffractive element. By blocking some portions of the microstructures, it modifies how light interacts with them and alters both amplitude and phase characteristics.
The shield effect is particularly relevant when dealing with complex dual-layer or multi-level diffractive structures in which multiple layers have different heights and materials. In such cases, accurate modeling techniques must consider these factors to predict more realistic performance parameters such as efficiency distribution across wavelengths and angles.
The effective microstructure height here is due to the occlusion caused by the shadow effect. Since some microstructures are occluded by the shadow, they do not participate in phase modulation, so we only use the remaining microstructure height as the effective height. To account for this shield effect accurately, we can derive the effective microstructure height using the shadow length from the preceding sections.
The phase delay in Eq. (5) can be written as
The diffraction efficiency can be expressed as,
We found that this correction works well when the first diffraction angles are less than zero, but when the first diffraction angle is more than zero, the error is large compared to the actual situation [18]. This is caused by the nonconformity of DLDOEs. When the incident angle is negative, the first diffraction angle is more than zero, and the diffraction angle increases further, making the shadow effect insufficient to calculate the diffraction efficiency loss caused by this process. Therefore, we introduce the oblique factor parameter in the part where the first diffraction angle is greater than zero to correct the diffraction efficiency calculation and obtain satisfactory results.
2.4 Oblique factor of DLDOEs
The obliquity factor accounts for the deviation in diffraction behavior when light waves are incident at an angle other than normal to a diffractive structure. When light waves encounter a diffractive element at an oblique angle, they undergo both diffraction and refraction. The oblique factor considers this angular dependence by modifying the traditional scalar diffraction theory.
The calculation of the diffraction angle of light leaving the first-layer diffractive optical element is rather complicated, because the ratio of the period width and microstructure height of the dual-layer diffractive optical element is small, making it difficult to use the existing scalar diffraction theory to calculate the diffraction efficiency distribution, especially when the diffraction efficiency does not meet 100%. Moreover, the diffraction efficiency distribution after passing through the first layer diffractive optical elements is not concentrated in a certain order, but is mainly distributed in two to three orders. After the second diffraction by the second-layer diffractive optical element, most of the light is finally diffracted into the same order (usually the first order). It is acceptable to use the refraction angle to approximate the diffraction angle corresponding to the maximum diffraction efficiency, because the gap between the two layers of diffractive optical elements is very small, and the period width of the elements is much larger than the wavelength, Therefore, we use the refraction angle to replace the diffraction angle, which can satisfy the calculation requirements, but note that the two angles are not exactly equal, which is also the main cause of the error of this method at large angles.
The diffraction angle of light passing through the first-layer diffractive optical element can be approximated as
Because the diffraction angle of light leaving the second layer diffractive optical element corresponds to the first-order diffraction, the angle is fixed. It can be obtained from the diffraction equation,
The obliquity factor is typically represented as K(θ), where θ is the angle of incidence. It quantifies the extent to which the efficiency or intensity distribution of diffracted light changes with respect to variations in θ. This correction can be significant when dealing with DLDOEs with larger angles of incidence or highly non-planar surfaces.
The obliquity factor of the diffractive optical element at each working wavelength is calculated using the following formula:
Based on the Kirchhoff’s diffraction formula,
The diffraction efficiency when the first diffraction angle is greater than zero can be expressed as,
As this method derives from the EAM method and the oblique factors, we name it the EAMK method. The ‘K’ in this case represents the inclusion of oblique factors.
The final diffraction efficiency over the entire incidence angle range can be expressed as
By incorporating oblique factors into the original EAM approach, we enhance its accuracy and applicability for diffractive elements under non-normal incidence conditions. This modified methodology can help account for deviations caused by incident angles and provide more precise predictions of diffraction efficiency distributions.
Based on the diffraction efficiency values under various incident wavelengths and incident angles, we can derive diverse parameters to assess the optical performance of DLDOEs, such as polychromic integral diffraction efficiency (PIDE) or multi-incidence angle polychromic integral diffraction efficiency (MPIDE).
The former can measure the mean diffraction efficiency of the diffraction element across the entire working band, and the latter can estimate the mean diffraction efficiency over the entire field of view and the entire band. We can use Eq. (18) as the objective function to optimize the microstructure height of the diffractive elements, so that they can obtain higher diffraction efficiency in the whole wavelength and field of view. We can also design other parameters according to specific situations to evaluate the diffraction elements, and adjust factors such as the material composition and microstructure height of the diffraction element to optimize these parameters to enhance the diffraction elements.
This paper is devoted to the improvement of the EAM method and its transformation into the EAMK method.
First, we improve the EAM method by simplifying the calculation of the shadow length. Instead of discussing the specific size of the incidence angle, we can calculate the shadow length by directly comparing the diffraction angle and the incidence angle.
Second, we introduce the oblique factor into the calculation of the diffraction efficiency, because when the incidence angle is negative, the main diffraction order deviates from the optical axis significantly, and the light in this case no longer satisfies the paraxial condition.
These two improvements together constitute the new EAMK method. The first improvement makes the EAMK method more concise, and the second improvement makes the EAMK method more accurate.
3. Simulations and results
In this section, we modeled a diffraction element with a finite periodic width to demonstrate the effectiveness of the EAMK method in analyzing the diffraction efficiency of DLDOEs under broad bandwidth and large incidence angle scenarios, computed its diffraction efficiency using scalar diffraction theory, EAM and EAMK method, and ultimately validated it using the time domain finite difference (FDTD) method.
For the working spectral range, based on the limit of the visible spectrum, we consider the edge values of the wavelengths as follows: λmin = 0.4 µm and λmax = 0.7 µm, the edge values of the incident angles as follows: θmin = -20°and θmax = 20°.
The first-layer HDE uses polymethyl methacrylate (PMMA) material, and the second-layer HDE uses polycarbonate (PC) material. The choice of these two materials is only for the convenience of calculation and illustration, and there is no special intention. The conclusion of this study does not depend on the choice of materials.
The microstructure height mainly determines the diffraction efficiency of the diffraction element, and the incident light wavelength, incident angle, and period width also influence it. This part primarily quantifies the impact of incident light wavelength and incident angle on diffraction efficiency under a certain period, so the microstructure height used in this paper is obtained by EAM under normal incidence. These microstructure heights guarantee that the diffraction element achieves high diffraction efficiency under normal incidence.
The optimized microstructure height is 4.475 µm and 2.862 µm when the DLDOEs have a period width of 20 µm. The optimized microstructure height is 15.25 µm and 11.85 µm when the DLDOEs have a period width of 100 µm.
3.1 Effect of incident wavelengths
The diffraction efficiency of DLDOEs is typically highest when the incident angle of light is normal. The decrease in the diffraction efficiency is more pronounced for larger deviations from zero. Figure 2 compares the diffraction efficiency of three methods (EAMK, SDT, and FDTD) for different incident angles and wavelengths. The FDTD method is considered as the standard reference because it is based on the numerical solution of Maxwell’s equations. It can be seen that the EAMK method agrees well with the FDTD method in most cases, especially for small incident angles and long wavelengths.
Fig. 2. Relationship between diffraction efficiency and wavelengths for three methods when the period width is 20 µm. (a) Incident angle is -20°. (b) Incident angle is -10°. (c) Incident angle is 0°. (d) Incident angle is 20°. (e) Incident angle is 10°.
Figure 2(a) and (b) show the diffraction efficiency versus wavelength for incident angles of -20° and -10°. It can be seen that the EAMK and FDTD methods have very close curves, whereas the SDT method has a large deviation. Figure 2(c) shows the diffraction efficiency versus wavelength for an incident angle of 0°. It can be seen that the SDT and EAMK have relatively close curves, but the EAMK is still more accurate.
Figure 2(d) and Fig. 2(e) show the diffraction efficiency versus wavelength for incident angle of 20° and 10°. SDT disregards the sign of the incident angle when computing the diffraction efficiency distribution, so the diffraction efficiency distributions computed by SDT in Fig. 2(a) and Fig. 2(d), Fig. 2(b) and Fig. 2(e) are identical, but the simulation results of FDTD indicate that when the incident angle varies from positive to negative directions, the diffraction efficiency distribution differs. The results derived by EAMK analysis agree with FDTD method, and exhibit similar change patterns.
Fig. 3. Relationship between diffraction efficiency and wavelengths for three methods when the period width is 100 µm. (a) Incident angle is -20°. (b) Incident angle is -10°. (c) Incident angle is 0°. (d) Incident angle is 20°. (e) Incident angle is 10°.
When the period width of DLDOEs increases to 100 µm, the optimized microstructure heights are 15.25 µm and 11.85 µm, and the diffraction efficiency distribution at different wavelengths is shown in Fig. 3. Figure 3(a) to (e) show the diffraction efficiency as a function of wavelength for five incidence angles. It can be seen that although the diffraction efficiency distribution at different incidence angles differs from that of DLDOE with a period of 20 µm, the diffraction efficiency calculated by EAMK method is similar to that calculated by the FDTD method, with a smaller error than the SDT method. Moreover, compared to Fig. 2, the error of EAMK method also decreases synchronously as the period width increases. This indicates that EAMK method can be used to analyze DLDOEs with different period widths, especially for DLDOEs with large periods and large incidence angles, the EAMK method has a great advantage. Because when the period and incidence angle increase, the FDTD method requires a lot of computational resources. The SDT method cannot accurately calculate the diffraction efficiency under large angle incidence.
Fig. 4. Comparison of PIDE and symmetry for four methods (EAMK, EAM, SDT, and FDTD) of DLDOE when the period width is 20 µm.
The SDT method shows larger deviations than the FDTD method, especially for large incident angles and short wavelengths. This indicates that the EAMK method is more accurate and reliable than the SDT method for calculating the diffraction efficiency of this grating structure.
3.2 Effect of incident angles
Figure 4 depicts the PIDE as a function of the incidence angles of DLDOEs when the period width is 20 $\mathrm{\mu}$m. Four curves are plotted in the figure: a blue solid line for EAMK, a red dashed line for SDT, and a yellow dashed line for FDTD. The blue curve has the highest PIDE values, followed by the red curve and the yellow curve.
As shown in Fig. 4, the EAMK method further enhances the accuracy of the EAM in calculating the diffraction efficiency, especially when the incident angle is negative. EAMK and FDTD have similar computational results. In contrast, the diffraction efficiency calculated by SDT at large incidence angles differs greatly from the rigorous vector theory. The difference between the SDT results and the rigorous vector theory implies that trusting the SDT calculations alone may lead to discrepancies between the expected and actual diffraction efficiencies of the fabricated diffraction elements. This disparity potentially results in failures in achieving the desired performance of the DLDOE.
Table 1 gives the PIDE values and errors at different incidence angles for three methods of DLDOE when the period width is 20 µm.
Table 1. PIDE values and errors for three methods of DLDOEs when the period width is 20 µm
The PIDE described by EAMK is consistent with the actual situation as a function of the incidence angle, with an error within 7.6%. The PIDE of DLDOE reaches its maximum value at an incidence angle of 3°, and decreases sharply as the incidence angle deviates from 3°. At an incidence angle of -20°, the PIDE calculated by FDTD is 36%, the PIDE calculated by EAMK is 41.52%, and the error is 5.47%. The PIDE calculated by EAM is 67.03%, and the error is 30.98%. Compared to the EAM method, the EAMK method increased the accuracy by 25.21%. The diffraction efficiency calculated by SDT is 91.14%, and the error is 55.09%.
The error of the EAMK and SDT method varies with the increase of the incidence angle, but the error of the EAMK method is always smaller than that of the SDT method. This indicates that the EAMK method is an accurate method that can maintain a small difference with the FDTD method, while the SDT method is an inaccurate method that has a large difference with the FDTD method.
The EAMK method and the SDT method have the smallest error near 0°, and the largest error near ±20°. This indicates that the EAMK method and the SDT method perform well at normal incidence, but poorly at oblique incidence.
Notably, when the incidence angle is negative, PIDE decreases much faster than when it is positive. This information can serve as a reference for the design of DLDOEs. To maximize diffraction efficiency when using DLDOEs, it is advisable to have the light incident from a positive angle. When the incidence angle is 20°, the PIDE can reach 71%, which is 35% higher than that at -20°.
Figure 5 shows the relationship between the PIDE and the incidence angle when the period width of DLDOEs increases to 100 µm. It can be seen that as the period width increases, the error of EAMK and FDTD further decreases. The error of SDT also decreases, but there is still a large gap between the results of the other two methods, especially in the case of the negative incidence angles. When the incidence angle is 1°, the PIDE reaches the maximum value, and the values calculated by the four methods are close at this angle, which indicates that SDT is accurate at small angles, but as the incidence angle exceeds 10°, the error reaches 13%, especially when the incidence angle is less than -10°, which reaches 23%. To more accurately analyze the accuracy of the three methods, Table 2 lists the PIDE values and corresponding errors calculated by the four methods at five incidence angles of -20, -10, 1, 10, and 20. Within ±20 degrees range, the EAMK error is within 5.26%, while the SDT error increases sharply as the absolute value of the incidence angle increases, reaching 65% at -20°. Compared to the EAM method, the EAMK method increased the accuracy by 21.53% when the incident angle is -20°.
Table 2. PIDE values and errors for three methods of DLDOE when the period width is 100 µm
In summary, the EAMK method is a superior method that can be used for the design and analysis of diffraction optical elements. In contrast, the SDT method is an inferior method that is not suitable for the design and analysis of diffraction optical elements.
3.3 Effect of microstructure period widths
To assess the applicability of the EAMK and SDT methods, it is essential to investigate the MPIDE values for different period widths, to estimate the average diffraction efficiency over the entire field of view and working band.
Figure 6 depicts the MPIDE of DLDOE as a function of the diffraction microstructure period width. The microstructure heights are constant at 15.25 µm and 11.85 µm, regardless of the period widths.
The diffraction efficiency calculation of SDT only depends on the microstructure height, and does not consider the period width parameter. Therefore, it is shown as a horizontal line in Fig. 6. As the period width increases, the shadow length and occlusion length decrease accordingly, and the MPIDE predicted by EAMK also increases sharply, reaching the same value as SDT. This indicates that for large period DLDOE, SDT can calculate accurately, but for small-period devices, additional corrections need to be introduced. With the help of EAMK method, the diffraction efficiency at different wavelengths and incident angles can be calculated simply and accurately. For the DLDOE in this example, when the period width is greater than 145 µm, the error of SDT is within 5%. When the period width is greater than 500 µm, the error of SDT is reduced to 3%.
4. Conclusions
In this study, we propose a simple and efficient method that combines the effective area method and the oblique factor to correct the SDT results for DLDOEs at large incident angles. The method can overcome the limitations of the SDT method and provide results comparable to the rigorous vector methods. We validate their proposed method by comparing it with FDTD and SDT methods using numerical simulations. we use DLDOEs with period lengths of 20 µm and 100 µm as examples, and analyze their diffraction efficiency distribution at different wavelengths (from 400 nm to 700 nm) and incident angles (from -20° to 20°). Our method can achieve results close to the FDTD methods, while the SDT method deviates significantly from reality as the incident angle increases. The method can be used to optimize the design of DLDOEs by adjusting their microstructure parameters, such as period width, microstructure height, and blazed angle.
Our method provides a simple and efficient way to design and apply DLDOEs for various optical functions and applications. Our work contributes to the advancement of diffractive optics and optical engineering, as it addresses an important problem in the field and provides a new solution that can improve the performance and functionality of DLDOEs. Our work also has potential applications in various fields such as optical communication, optical computing, optical imaging, and optical sensing.
Compared to existing methods, our method offers advantages in simplicity, accuracy, and versatility. It can be readily implemented and extended to multi-layer diffractive optical elements with more complex structures and functions. Moreover, it can handle various incident angles and wavelengths, making it adaptable to different scenarios and requirements. However, the EAMK method has its limitations. It simplifies the computation by approximating the diffraction angle with the refraction angle. This approximation is only valid when the diffraction angle of the diffraction element is similar to the refraction angle. If the incident angle exceeds 20° (absolute value), a significant error occurs. Nevertheless, it is worth noting that the diffraction efficiency of the diffraction element decreases significantly at such large angles, which is why multi-layer diffraction elements are generally not used in these cases. Additionally, it is important to mention that this method is based on scalar diffraction theory and is only suitable for multi-layer diffractive optical elements with a period width greater than 20 times the wavelength.
We hope that our paper will inspire further research and development on DLDOEs and other diffractive optical elements, as well as their applications in various domains. We also hope that our paper will provide useful guidance and reference for researchers and engineers who are interested in or working on DLDOEs and related topics.
Funding
Natural Science Foundation of Shandong Province (ZR2020QF099); National Natural Science Foundation of China (62205238).
Disclosures
The author declares no conflicts of interest.
Data availability
No data were generated or analyzed in the presented research.
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