Open Access
Add to BibSonomy
Abstract
For a waveguide display device, the field of view (FOV) is a key parameter for evaluating its optical performance. To address this issue, we propose a hybrid waveguide system, which is composed of two projectors, two in-couplers, two half-mirror arrays and an out-coupler. We use two projectors to generate the left and right parts of the output image separately, which can increase the upper limit of the FOV significantly. Unlike conventional waveguide-based system, we use half-mirror arrays instead of folding gratings to realize 2D exit pupil expansion. By doing so, the total internal reflection condition can always be met during the pupil expansion process. To solve the difficulty in designing collimating optical system with large FOV, we propose a method of tilting the projection system. The hybrid waveguide system can realize a FOV of 88°(H) × 53°(V).
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Recent years, augmented reality (AR) technology has been attracting more and more attention. Through AR display devices, we can see the virtual images and the real-world scene at the same time, which can improve the efficiency of obtaining information greatly. AR technology has been widely used in military, entertainment, medical treatment and education [1–3].
There are various approaches to realize AR, including half-mirror [4–6], freeform optical prism [7,8], retina scanning [9,10], geometrical waveguide [11,12] and diffractive waveguide [13–15]. Among these approaches, AR display devices based on optical waveguide have the advantages of compactness, wide field of view (FOV) and large eye box, which can provide good user experiences. For AR display system, the field of view is a key parameter for evaluating its optical performance, and expanding the FOV of AR devices has always been the research hotspot. As to waveguide-based system, the FOV is mainly limited by the total internal reflection (TIR) condition of the waveguide. In order to break the limitation of TIR condition, some multi-channel designs with large FOV have been proposed. Saarikko et al. [16]. proposed a double-layered grating waveguide system, in which light with different incident angles is coupled into different layer and propagates separately, and the horizontal FOV can be doubled. Wang et al. [17] proposed a double layered geometrical waveguide system. In this design, two projectors are used to generate light of different fields, and light from each projector is injected into corresponding waveguide layer. However, 2D exit pupil expansion (EPE) can not be realized in the designs above. For waveguide-based display system with large FOV, the exit pupil will be very small if without 2D EPE, and the displayed image may be incomplete when the eyeball rotates. Double layered structure will also increase the weight and volume of the device, and the transmission efficiency of the external light will decrease. Han et al. [18] proposed a method for achieving FOV expansion within a single layer of waveguide. Light with different incident angles propagates in different regions within the waveguide. However, 2D EPE is not realized in this design. Lushnikov et al. [19] uses 2D grating as in-coupler, which can separate light of different fields, and 2D EPE can also be achieved. However, The TIR condition of the corner field will be broken during EPE process, and the output image will be incomplete. Chen et al. [20] put two folding gratings at both sides of the in-coupler, which can separate the light of different fields and realize 2D EPE. Though the horizontal FOV can be expanded significantly, the vertical FOV is still limited in this design.
In this paper, we propose a hybrid waveguide system that can provide large FOV both horizontally and vertically. And effective 2D EPE can also be realized. In our design, we use two projectors to generate left and right halves of the output image, and there are two in-couplers in the waveguide accordingly. Unlike conventional grating waveguide system, we use half-mirror arrays rather than folding gratings to realize 2D EPE, which can make the TIR condition always be met during the pupil expansion process. The FOV of the hybrid waveguide system is 88°(H) × 53°(V), and the corresponding diagonal FOV is 94.7°. Two surface-relief gratings are designed as the in-coupling and out-coupling gratings based on rigorous coupled wave analysis. The diameter of the exit pupil is 10 mm at an eye relief of 10 mm. To solve difficulty in designing collimating optical system with large FOV, we propose a method of tilting the projection system, and the required FOV for the collimating lens has been reduced significantly.
2. Principle
2.1 Wave vector domain analysis
For a waveguide-based display system, the FOV is mainly limited by the TIR condition of the waveguide. The angular range of the incident light that can propagate in the waveguide losslessly after in-coupling can be well described in the wave vector domain (k-domain) [21]. Assuming that the virtual image is rectangular, the definitions of horizontal FOV ${\theta _H}$ and vertical FOV ${\theta _V}$ are shown in Fig. 1. The schematic of a grating waveguide system with 1D EPE is shown in Fig. 2(a). Incident light is coupled into the waveguide by the in-coupling grating. Then the out-coupling grating with the same period couples the light out of the waveguide and expands the exit pupil horizontally. In the in-coupling diffraction, the diffraction angle of the first reflective order should meet the TIR condition. The TIR condition in the k-domain can be expressed as:
where ${n_w}$ is the refractive index of waveguide, and the parameters α and β can be expressed as:In order to break the limitation of the TIR condition, some designs with two channels have been proposed. For example, large FOV can be realized by a double-layered waveguide [16]. The basic schematic is shown in Fig. 3(a), and the k-domain analysis diagram is shown in Fig. 3(b). The field of view is split into two halves, and the light of each sub field propagates within its corresponding waveguide. The grating periods of the two waveguides are different, which makes the TIR condition be met for both sub fields. In the k-domain diagram, different grating period means different shift distance for the diffraction process. Appropriate grating periods can make the two parts of the curved rectangle fall within the annular region after in-coupling. Theoretically, the maximum horizontal FOV of the double-layered waveguide system can be expressed as:
Fig. 2. General 1D EPE grating waveguide system. (a) Basic schematic of waveguide system. (b) K-domain analysis diagram of 1D EPE system.
Fig. 3. Principle of double-layered grating waveguide system. (a) Basic schematic of optical path (WG is for waveguide). (b) K-domain analysis diagram.
We can see that the FOV can be expanded significantly by splitting the field of view. However, if 2D EPE is not realized, it is difficult to achieve good user experience.
By adding extra grating, it is possible to achieve 2D EPE. A design that can provide large FOV as well as 2D EPE has been proposed [20], and the basic schematic is shown in Fig. 4(a). In this design, the in-coupling grating can guide the incident light into the waveguide towards both left and right directions. Then the light is redirected by two folding gratings, and leaks out from the waveguide at the out-coupling grating. The analysis diagram in wave vector domain is visualized in Fig. 4(b). Field 1 and Field 2 represent left sub field and right sub field, respectively. Though the horizontal FOV can be expanded significantly in this design, the vertical FOV is still seriously limited by the TIR condition. If the vertical FOV is too large, TIR condition will be broken after redirection process. The upper limits of the horizontal and vertical FOV of this system can be expressed as:
Fig. 4. Principle of large FOV grating waveguide system with 2D EPE. (a) Waveguide layout. (b) K-domain analysis diagram.
2.2 Hybrid waveguide
For a waveguide-based display system with large FOV in two dimensions, 2D EPE is needed for good user experience. To address this challenge, we propose a hybrid waveguide system that uses half-mirror array, which is usually used in geometrical waveguide system, instead of folding grating to realize 2D EPE. The basic schematic of the hybrid waveguide is shown in Fig. 5(a). There are two in-coupling gratings in the waveguide. And two projectors are used to produce the left and right parts of the virtual image, respectively. The sectional view of the collimating and in-coupling system is shown in Fig. 5(b). Figure 5(c) shows the basic schematic of 2D EPE, and the model is built in Lighttools software. After being coupled into the waveguide, the light is redirected by the half-mirror array and pupil expansion along y axis can be realized by multiple reflections. All half-mirrors are perpendicular to the waveguide. The reflected light leaks out from the waveguide at the out-coupling grating and 2D EPE is realized.
Fig. 5. Basic schematic of hybrid waveguide system. (a) Front view of waveguide layout. (b) Sectional view of collimating and in-coupling system. (c) Basic schematic of 2D EPE (d) K-domain analysis diagram.
The changing process of wave vector in the hybrid waveguide is visualized in Fig. 5(d). In the wave vector domain, half of the rectangular virtual image can be described as a curved rectangle with one straight edge. The curved rectangle that represents the incident light is located below the horizontal axis. After being diffracted by the in-coupling grating, the curved rectangle shifts downwards to the lower side of the annular region. In our design, the angle between half-mirrors and y axis is 45°. For the light from the left projector, if the k-vector is $({{k_x},{k_y},{k_z}} )$ before reflection, the k-vector will become $({ - {k_y}, - {k_x},{k_z}} )$ after being reflected by the half-mirror arrays. Thus, the curved rectangle will flip rather than shift. For the light from the left projector, the curved rectangle will move to the right side of annular region after reflection by the left half-mirror array. The same goes for the light from the right projector. The curved rectangle will not leave the annular region through flipping. In other words, light that meets the TIR condition in the waveguide can still meet the TIR condition after being reflected by a mirror in waveguide if the mirror is perpendicular to the waveguide. Therefore, the problem that the vertical FOV is limited by the TIR condition can be solved by using half-mirror array instead of folding grating. After being diffracted by out-coupling grating, two curved rectangles move to the center of k-domain diagram and the TIR condition is broken. Light from two image sources can be combined to form a complete image with a large FOV. For a general grating waveguide system, the k-vectors in the XY plane before in-coupling and after out-coupling are the same, and the transformation can be expressed as $({{k_x},{k_y}} )$ to $({{k_x},{k_y}} )$. after adding mirrors, for the light from the right projector, the transformation can be expressed as $({{k_x},{k_y}} )$ to $({{k_y},{k_x}} )$. And the transformation can be expressed as $({{k_x},{k_y}} )$ to $({ - {k_y}, - {k_x}} )$ for the light from the left projector. Though the original k-vector is changed, one input k-vector still have one corresponding output k-vector. Though EPE methods based on both reflection is diffraction are used, the dispersion will not happen.
The upper limit of the horizontal FOV of this system can be expressed as:
When the FOV grows, the curvature radius of the curved rectangular that represents FOV in the k-domain diagram will gradually decrease. For a symmetric FOV, the curvature radius of the left and right sides of the curved rectangle can be expressed as:
When the horizontal FOV reaches the upper limit of the waveguide, the curvature radius is:
If the refractive index of the waveguide is larger than 1.618, the curvature radius of left and right sides of the curved rectangular will be smaller than the outer radius of TIR region, which means arbitrary vertical FOV can be achieved. Assuming that ${n_w}$ is 1.8, the maximal horizontal FOV is 106.3°. In k-domain, the curved rectangles that represent half of the virtual images with different vertical FOV is shown in Fig. 6. We can see that the curved rectangles can always fall within the annular region when the vertical FOV grows. Considering the aspect ratio of the image sources and the sight range of human eye, the designed vertical FOV will not be very large.
Fig. 6. The range of wave vector with different vertical field of view. The refractive index of waveguide is 1.8, and the horizontal field of view is 106.3°
3. Optical system design
3.1 Collimating system design
In our design, the wavelength is 532 nm and the grating period T of the in-couplers and the out-coupler and is set as 530 nm. We choose the HOYA-FD60W glass as the waveguide material considering its high refractive index of 1.817 at 532 nm. The upper limit of the horizontal FOV is 108.8° if we only consider the TIR condition. However, if the diffraction angle in the waveguide is very large, the distance between adjacent replicated exit pupil will be large and dark area may appear when light from certain viewing angles can not enter the human eye. Generally, the diameter of human eye pupil indoors is about 4 mm. If the diffraction angle is 70 degrees, the distance of two adjacent output beams is 5.495 mm. As the diameter of the in-coupler is 2 mm, the gap between two output beams is 3.495 mm. If we set the maximal diffraction angle at 70 degrees, light of all fields can enter the eye for indoor scenes. Horizontal FOV of 88 ° can meet the requirement.
If we design the collimating optical system as Fig. 5(b), in which the optical axis is perpendicular to the waveguide, the FOV of the collimating lens will be equal to the FOV of the whole system, which can result in high difficulty in designing collimating lens, and half of the field will not be utilized. To address this issue, we consider rotating the projection system and making the image source coaxial to the collimating lens. The basic schematic of the tilted projection system is shown in Fig. 7(a). The required horizontal FOV for each projector is ${{{\theta _H}} / 2}$, where ${\theta _H}$ is the horizontal FOV of the whole system. When the two sub fields can be perfectly aligned in the horizontal direction, the rotation angle of the projection system can be expressed as:
Based on the design above, we use image sources with an aspect ratio of 4:3 and the image sources are arranged vertically. If the distortion of the collimating system is neglected, the full size of output image is shown in Fig. 7(b). If we choose the maximal rectangle to be the output image, the vertical FOV is 53° and the corresponding diagonal FOV is 94.7°. The required FOV for the collimating system is 67.9°, which has been significantly reduced compared with the FOV of the whole system. Due to the tilting of the projection system, the displayed image will be deformed, and the problem can be solved through image preprocessing.
Fig. 7. (a) Basic schematic of tilted projection system. (b) Full size of output image generated from two image sources.
Based on the calculation results above, a collimating lens is designed. In order to use more energy of the image sources, the chief ray of each field is closely perpendicular to the image source. The optical layout is shown in Fig. 8(a). The focal length is 6 mm, and the pupil diameter is 2 mm. We choose a 0.27-inch micro-display with 1024 × 768 pixels as the image sources, and the corresponding Nyquist frequency is 94 lp/mm. The modulation transfer function (MTF) is shown in Fig. 8(b), and all MTF values are above 0.372 at 94lp/mm, which is sufficient for the visual system. The distortion curve is shown in Fig. 8(c), and the maximal distortion is 20.2%.
Fig. 8. Design result of the collimating lens. (a) Layout and ray-tracing. (b) MTF curve. (c) Distortion curve.
3.2 Gratings design
The grating structures of the in-coupler and out-coupler are illustrated in Fig. 9. The in-coupling grating with a sawtooth structure can achieve high diffraction efficiency on the first reflective order (R1). The diffraction efficiency can be further improved by coating a layer of TiO2 and a layer of sliver. We choose nanoimprint polymer that refractive index is 1.76 at 532 nm. The out-coupling grating has a symmetrical structure, so that the light from both sides has the same diffraction efficiency. For the out-coupling grating, the energy should concentrate on the zeroth reflective order (R0) as well as the minus first reflective order (R−1)
We use rigorous coupled wave analysis (RCWA) to optimize the parameters of the gratings. The average diffraction efficiency and the efficiency uniformity are considered in our optimization. The optimized parameters of the gratings are summarized in Table 1. The blaze angle of the in-coupling grating is 19.72°. The diffraction efficiencies of the two gratings when the incident angle changes are shown in Fig. 10. The mass manufacturing of surface relief gratings can be realized by nanoimprint technology [22]. The grating master can be made in silicon by electron beam lithography (EBL) and dry eching [23], and the grating structure can be transferred to the waveguides at high accuracy through nanoimprint with low cost.
Fig. 10. Diffraction efficiencies of (a) in-coupling grating and (b) out-coupling grating. TE for transverse electric mode and TM for transverse magnetic.
Table 1. Optimized parameters and average diffraction efficiencies of gratings
3.3 Waveguide design
Appropriate exit pupil size and eye relief are important for the user experience, which has requirements for the size of the out-coupling grating. For example, in the vertical direction, as shown in Fig. 11(a), the beams of the fields $({0,{{{\theta_V}} / 2}} )$ and $({0,{{ - {\theta_V}} / 2}} )$ are required to be out-coupled at corresponding regions. The size of the out-coupling regions are the same as the exit pupil. The requirements for the horizontal size OGX and the vertical size OGY of the out-coupling grating can be expressed as:
Fig. 11. (a) Schematic diagram of exit pupil in y-z view. Red parallelogram for the light path of the field (0,θV/2) and green parallelogram for the light path of the field (0,-θV/2). (b) Schematic diagram of the position relationship between the half-mirror arrays and the out-coupling grating in x-y view. Yellow regions for half-mirror arrays, blue rectangle for out-coupling grating, red circle for out-coupling area of the field (0,θV/2) and green circle for out-coupling area of the field (0,-θV/2). (c) Schematic diagram of the shift of the light in field (0,-θV/2) through multiple reflections.
To ensure that the light of each field can exit the waveguide at a specific area of the out-coupling grating and pass through the exit pupil, some requirements for the size of the half-mirror arrays need to be met. For the light in the waveguide, we define the k-vector along x axis and y axis as ${k_x}$ and ${k_x}$, respectively, and the maximal incline angle in the XY plane can be expressed as:
In our design, LM is 33.4 mm. For the light from the left projector, the propagation direction of the field $({0,{{ - {\theta_V}} / 2}} )$ is towards the lower left after in-coupling, and the light can not propagate to the lower right corner of the half-mirror array directly. However, as shown in Fig. 11(c), the light can be reflected multiple times by the half-mirrors and shift to the right. Thus, the light of field $({0,{{ - {\theta_V}} / 2}} )$ can propagate to the lower right corner through the half-mirror array.
Based on the calculation above, the designed 2D layout of the hybrid waveguide is shown in Fig. 12(a). In order to fully utilize the energy of the coupled light, the shape of the half-mirror arrays are right trapezoids, and the bevel angle is φ. Instead of using rectangular shape, trapezoid can make the size of the waveguide smaller in commercial products without wasting energy. The in-coupling gratings are circles with a diameter of 2 mm, which can match the pupil of the projection system. The thickness of waveguide is 1 mm and the distance between two adjacent half-mirrors is 1.414 mm. The vertical distance between two half-mirrors is 2 mm, which can provide good EPE effect in the vertical direction. There are 27 half-mirrors in each array. The reflectivity of the first 17 half-mirrors is 0.1, and the reflectivity of the last 10 half-mirrors is 0.25. 3D model of the whole system based on Lighttools software is shown in Fig. 12(b). Here we present a possible solution for the actual production. The gratings can be manufactured on a piece of very thin glass by nanoimprint. Then the hybrid waveguide can be manufactured by stacking the thin glass to a piece of geometrical waveguide [24].
Fig. 12. Design result of hybrid waveguide AR display system. (a) 2D layout of hybrid waveguide. Gray circles for in-coupling gratings, blue rectangle for out-coupling grating and oblique black lines for half mirrors. (b) 3D view of whole system in Lighttools software.
We use the images shown in Fig. 13(a) and Fig. 13(b) as the input images of the left and right image sources, respectively, and we add a perfect lens to obtain the output image. The image at the focal plane is shown in Fig. 13(c). We can see that the system can present the input image clearly. We use the picture of grid shown in Fig. 14(a) as the input image for both image sources, and the output image is shown in Fig. 14(b). The red rectangle in Fig. 14(b) represents the FOV of 88°×53°. We can see that the target FOV can be realized in our design. Due to the tilt of the collimating system and the distortion of the lens itself, there are distortion in the output image. The relative illumination distribution at the image plane within the full FOV is shown in Fig. 15(a). The overall efficiency of the waveguide system with different horizontal field angles is shown in Fig. 15(b). Note that due to the cosine-fourth-power law [25], the illumination distribution in the image plane is not proportionate to the efficiency of the waveguide. The efficiency is the proportion of collected energy at the exit pupil to the incident energy. The problems of distortion and angular nonuniformity can be solved through image preprocessing.
Fig. 13. Simulation result. (a) Input image of left image source. (b) Input image of right source. (c) Output image focused by a perfect lens.
Fig. 14. (a) Input image for both image sources. (b) Output image. The red rectangle represents the FOV of 88°×53°.
Fig. 15. (a) Relative illumination distribution at the image plane within the full FOV. (b) Overall efficiency of the waveguide system when the horizontal field angle changes.
4. Conclusions and discussions
For an augmented reality display system, the FOV is one of the most important parameters for evaluating its optical performance, and researchers have been struggling to expand it. The FOV of waveguide-based system is mainly limited by the TIR condition. Though many multi-channel designs which can break the limitation have been proposed, they still have some defects. Some of them are not able to realize effective 2D EPE, which is vital for user experience. And some of them can only expand FOV on one dimension.
In this paper, we systematically analyze the performances of previous works in wave vector domain and propose a hybrid waveguide system that can provide large FOV both horizontally and vertically, as well as 2D EPE. In our design, we use two projectors to generate the left and right parts of the output image separately. We use half-mirror arrays rather than folding gratings to realize 2D EPE. The FOV of our design is 88°(H) × 53°(V), and the corresponding diagonal FOV is 94.7°. Based on RCWA, two diffraction gratings are designed as the in-coupling and the out-coupling gratings. The diameter of the exit pupil is 10 mm at an eye relief of 10 mm. To solve difficulty in designing collimating optical system with large FOV, we propose a scheme of tilting the projection system, and the required FOV for the collimating lens has been greatly reduced. This design gives an effective method to realize extra large FOV for waveguide-based system, which can enhance the user experience of AR devices greatly.
The improvement of the spatial uniformity for the hybrid waveguide system could be a challenging work. For general grating waveguide system, the spatial uniformity can be improved by changing the diffraction efficiency gradually. However in our design, the out-coupling grating is used for beams from two directions. This problem could be solved by designing asymmetric grating. However, the required algorithm for optimization is challenging. The optimization of the reflectivity distribution of the half-mirrors is also challenging due to multiple reflection effect. The distortion of our system is severe, and image preprocessing is not the best solution to address it. The freeform optics can provide more design freedom and better optical performance. Designing a collimating system based on freeform optics could be a possible solution to correct the distortion caused by the tilt of the collimating system.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. A. A. Cameron, “Optical waveguide technology and its application in head-mounted displays,” Head-and Helmet-Mounted Displays XVII; and Display Technologies and Applications for Defense, Security, and Avionics VI. Vol. 8383, SPIE, (2012).
2. T. Sielhorst, M. Feuerstein, and N. Navab, “Advanced medical displays: A literature review of augmented reality,” J. Disp. Technol. 4(4), 451–467 (2008). [CrossRef]
3. H. Kaufmann, K. Steinbügl, A. Dünser, and J. Glück, “General training of spatial abilities by geometry education in augmented reality,” Annual Review of CyberTherapy and Telemedicine: A Decade of VR 3, 65–76 (2005). [CrossRef]
4. M. Antonio, J. Bayo-Monton, A. Lizondo, C. Fernandez-Llatas, and V. Traver, “Evaluation of Google Glass technical limitations on their integration in medical systems,” Sensors 16(12), 2142 (2016). [CrossRef]
5. S. Ashok, A. Riser, and J. R. Rogers, “Design of an advanced helmet mounted display (AHMD),” Cockpit and Future Displays for Defense and Security. Vol. 5801. International Society for Optics and Photonics, 2005.
6. L. Wei, Y. Li, J. Jing, L. Feng, and J. Zhou, “Design and fabrication of a compact off-axis see-through head-mounted display using a freeform surface,” Opt. Express 26(7), 8550–8565 (2018). [CrossRef]
7. D. Cheng, Y. Wang, H. Hua, and M. M. Talha, “Design of an optical see-through head-mounted display with a low f-number and large field of view using a freeform prism,” Appl. Opt. 48(14), 2655–2668 (2009). [CrossRef]
8. D. Cheng, Y. Wang, H. Hua, and J. Sasian, “Design of a wide-angle, lightweight head-mounted display using free-form optics tiling,” Opt. Lett. 36(11), 2098–2100 (2011). [CrossRef]
9. J. Xiong, G. Tan, T. Zhan, and S. Wu, “A scanning waveguide AR display with 100° FOV,” Optical Architectures for Displays and Sensing in Augmented, Virtual, and Mixed Reality (AR, VR, MR) II. Vol. 11765. International Society for Optics and Photonics, 2021.
10. R. Gabai, G. Cahana, M. Yehiel, G. Yearim, T. Yusupov, A. Baram, and M. Naftali, “Enhanced 3D perception using Laser based scanning display,” Optical Architectures for Displays and Sensing in Augmented, Virtual, and Mixed Reality (AR, VR, MR) II. Vol. 11765. International Society for Optics and Photonics, 2021.
11. J. Yang, P. Twardowski, P. Gérard, and J. Fontaine, “Design of a large field-of-view see-through near to eye display with two geometrical waveguides,” Opt. Lett. 41(23), 5426–5429 (2016). [CrossRef]
12. K. Zhao and J. Pan, “Optical design for a see-through head-mounted display with high visibility,” Opt. Express 24(5), 4749–4760 (2016). [CrossRef]
13. Z. Liu, Y. Pang, C. Pan, and Z. Huang, “Design of a uniform-illumination binocular waveguide display with diffraction gratings and freeform optics,” Opt. Express 25(24), 30720–30731 (2017). [CrossRef]
14. J. Xiao, J. Liu, J. Han, and Y. Wang, “Design of achromatic surface microstructure for near-eye display with diffractive waveguide,” Opt. Commun. 452, 411–416 (2019). [CrossRef]
15. Y. Zhang and F. Fang, “Development of planar diffractive waveguides in optical see-through head-mounted displays,” Precis. Eng. 60, 482–496 (2019). [CrossRef]
16. S. Pasi, “Diffractive exit-pupil expander with a large field of view,” Photonics in Multimedia II. Vol. 7001. SPIE, 2008.
17. Q. Wang, D. Cheng, Q. Hou, L. Gu, and Y. Wang, “Design of an ultra-thin, wide-angle, stray-light-free near-eye display with a dual-layer geometrical waveguide,” Opt. Express 28(23), 35376–35394 (2020). [CrossRef]
18. W. Han, J. Jeon, M. Choi, and J. Park, “Lightguide type Maxwellian near-eye display with enlarged horizontal field of view by optical reconfiguration of input image,” J. Inf. Disp. 23(3), 201–211 (2022). [CrossRef]
19. D. S. Lushnikov, A. Y. Zherdev, S. B. Odinokov, V. V. Markin, M. V. Shishova, and A. B. Solomashenko, “Development and research of a waveguide for augmented reality systems with diffractive relief-phase gratings,” Optics, Photonics and Digital Technologies for Imaging Applications VI. Vol. 11353. SPIE, 2020.
20. C. Chen, L. Mi, W. Zhang, J. Ye, and G. Li, “Waveguide-based near-eye display with dual-channel exit pupil expander,” Displays 67, 101998 (2021). [CrossRef]
21. J. E. Harvey and C. L. Vernold, “Description of diffraction grating behavior in direction cosine space,” Appl. Opt. 37(34), 8158–8159 (1998). [CrossRef]
22. S. Steiner, M. Jotz, F. Bachhuber, B. Bilenberg, T. Bro, J. Meulen, E. Ercan, A. Rashed, and M. Deveci, “Enabling the Metaverse through mass manufacturing of industry-standard optical waveguide combiners,” Optical Architectures for Displays and Sensing in Augmented, Virtual, and Mixed Reality (AR, VR, MR) IV. Vol. 12449. SPIE, 2023.
23. Z. Liu, D. Wang, H. Gao, M. Li, H. Zhou, and C. Zhang, “Metasurface-enabled augmented reality display: a review,” Adv. Photonics 5(03), 034001 (2023). [CrossRef]
24. Z. Liu, C. Zhang, W. Zhu, Z. Huang, H. J. Lezec, A. Agrawal, and L. J. Guo, “Compact stereo waveguide display based on a unidirectional polarization-multiplexed metagrating in-coupler,” ACS Photonics 8(4), 1112–1119 (2021). [CrossRef]
25. C. Pan, Z. Liu, Y. Pang, X. Zheng, H. Cai, Y. Zhang, and Z. Huang, “Design of a high-performance in-coupling grating using differential evolution algorithm for waveguide display,” Opt. Express 26(20), 26646–26662 (2018). [CrossRef]
