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Abstract
Designing diffractive waveguides for head-mounted displays requires wide-angle conical diffraction analysis of multiple gratings. In this work, diffractive waveguide design using the relative direction cosine space, which extends the direction cosine space to a relative space involving refractive indices and can describe grating diffraction through various media, is demonstrated. A diffractive waveguide was fabricated with grating periods of 382 and 270 nm, which generated a monochromatic virtual image image in green light (520 nm). The maximum field of view was measured as 39° with 0.5° deviation from the center of view.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Head-mounted displays (HMDs) for augmented reality (AR) technologies have been undergoing rapid development and commercialization in recent years. For AR applications, an HMD mainly works as an optical combiner that couples the light from a virtual image with the environmental light and therefore allows the viewer to see the virtual image in a real-world sense [1–3]. Several optical solutions are available for HMDs, including geometric optical elements using a flat or a freeform beam splitter [4,5], geometrical waveguides adopting partially reflective or surface-relief mirror arrays [6–8], diffractive waveguides with surface-relief or holographic gratings [9–18], retinal scan displays [19], and computational holography [20–22]. Although these techniques offer advanced features, they compromise either optical performance or system compatibility and compactness [2,23].
Diffractive waveguides have recently attracted attention as one of the most favorable solutions for HMDs [9–18]. Figure 1 presents a conventional diffractive waveguide configuration with two-dimensional exit pupil expansion using three surface-relief gratings. The first grating (G1, in-coupling grating) couples the collimated light from a microdisplay into a substrate by diffraction and delivers the light toward the second grating (G2). G2 turns the light toward the out-coupling grating (G3). Meanwhile, the pupil is duplicated vertically (along the y axis in Fig. 1) by diffraction-induced beam splitting on G2. Then, G3 couples the light out of the waveguide toward the viewer and expands the exit pupil horizontally (along the x axis).
Although obtaining a large field of view (FOV), a wide eye box, and high-quality uniform images is challenging, diffractive waveguides offer several advantages; for example, they are lightweight, highly transparent, and suitable for mass production, especially in combination with surface-relief gratings fabricated by nanoimprint lithography [24–26]. Moreover, diffractive-waveguide-based HMDs improve the overall optical performance while maintaining the compactness of the system through an increased number of gratings or the integration of layouts with increased complexity. Due to the increasing demand for HMDs with high optical performance, the design of diffractive waveguides, especially the optimization of the gratings in the waveguide, is significant. The design process of a monochromatic diffractive waveguide can be divided into three phases:
- • Simultaneous determination of the periods, directions, and relative positions of the gratings to ensure that light out-coupled from the waveguide is at the same angle as the incident light in the widest angle range possible.
- • Optimization of the groove profiles of gratings to obtain high and uniform diffraction efficiency at various incident angles.
- • Optimization of the whole layout of the diffractive waveguide, including the position, shape, and size of each grating.
The second and third design phases can be achieved using rigorous coupled-wave analysis (RCWA) and commercial raytracing software [27,28], respectively, with an optimization algorithm such as a genetic algorithm to solve multiobjective optimization problems [29]. Softwares such as VirtualLab can even simultaneously address the second and third design phases including both near-field and far-field simulations [30]. However, calculating the diffraction angle of various incident angles through all the gratings inside the waveguide remains complicated for the first design phase. In addition, the collimated light from a two-dimensional image strikes the gratings in a different plane of incidence (nonparaxial diffraction) [31 32], which generates wide-angle conical diffraction in three-dimensional space and thus is much more complicated to analyze than paraxial diffraction, which is described by the well-known grating equation.
In this paper, we propose a relative direction cosine space (RDCS) to describe the wide-angle conical diffraction through multiple gratings, following Harvey and Vernold [32], where the shifts of the direction cosines caused by the grating diffraction do not rely on the refractive indices of the incident and exit media. Accordingly, the diffraction through multiple gratings in the diffractive waveguide is expressed by a series of shifts determined by the period of each grating. The optimal period and direction of each grating can thus be directly determined from a diagram of relative direction cosines for the maximum FOV of the diffractive waveguide, largely simplifying the design of the diffractive waveguide.
This article is organized as follows: the principle of the RDCS and RDCS diagram are demonstrated in Section 2. The diffraction behavior of an in-coupling grating is analyzed by the RDCS in Section 3. In Section 4, we prove that the optimal period and direction of each grating can be directly determined from the relative direction cosine diagram. Finally, the fabrication of a diffractive waveguide is demonstrated using the parameters optimized by our method. The validity of the optimization method is confirmed using a relative direction cosine diagram by observing a virtual image with a real-world sense through a diffractive waveguide.
2. Relative direction cosine diagram
To model the nonparaxial diffraction grating behavior of an arbitrary obliquely incident beam, Harvey and Vernold described the grating equation in terms of the direction cosines of the propagation vectors of the incident beam [32]. In the direction cosine space, wide-angle diffraction is expressed by shift invariance with respect to the incident angle; however, the shifts of the direction cosines depend on the refractive indices of the incident and exit media, and for a transmission-type grating with different refractive indices of the incident and exit media, the direction cosines of the diffracted light exhibit not only shifts but also scaling transformations from the direction cosines of incidence based on the refractive index radio between the incident and exit media. Although the diffraction of a single grating is concisely expressed, the direction cosine diagram is not suitable for being directly applied to analyze the diffraction behavior of multiple gratings with different refractive indices. Therefore, we proposed an RDCS that extends the direction cosine space to a relative space with different refractive index domains. The definition and derivation process of RDCS is shown as follows:
As the incident light of the diffractive waveguides is collimated, a wave vector is used in the medium $\vec{k}$ to represent the incident and far-field diffracted light. In Cartesian coordinates, $\vec{k}$ is expressed by a magnitude of 2π/λ and direction cosines (α0, β0, γ0) [32–35]:
The directions of incidence, diffracted light, and the grating can be defined by the polar angle θ and the azimuthal angle ϕ. As shown in Fig. 2 (a), the boundary plane is assumed on the x–y plane; the polar angle of the incident light θi indicates the angle from the + z axis to the wave vector, the azimuthal angle ϕi is the angle from the + x axis to the wave vector. Thus, the relative direction cosines can be express as
The grating direction is described by the grating vector $\vec{G}$, defined by
Fig. 2. (a) Demonstration of conical diffraction directions in Cartesian coordinates. (b) Conical diffraction and (c) shift of FOV described by the relative direction cosine diagram.
Diffraction equations using relative direction cosine are expressed by
where m = 0, ±1, ±2, ….; αm, βm, γm are the relative direction cosines of the mth-order diffracted light, and the superscripts T and R refer to transmission or reflection grating cases, respectively. Note that the diffraction equations (Eq. (5)–8) hold even when the grating vector ($\vec{G}$) is not parallel to the grating surface (e.g., volume holographic gratings). Furthermore, for a grating with two or more periods in different directions (e.g., two-dimensional gratings), there should be additive components on the right side of the equations to express the extra periods and directions.The diffraction equations imply an important feature of the relative direction cosines: the direction of the mth diffraction order of light through a grating can be expressed by shifting the incident relative direction cosines at a distance of mλ0/d along the corresponding grating vector. This feature is evident in the projection view of relative direction cosines on the α–β plane, which is also named as a relative direction cosine diagram, as shown in Fig. 2(b) and (c). Furthermore, according to Snell’s law, the reflected or refracted relative direction cosines α, β remain the same as the incident ones provided that the boundary plane is parallel to the x–y plane. This feature offers great convenience when coping with light propagation behavior between gratings.
In Fig. 2(b), various orders of the conical diffraction are depicted by red dots or circles, which are equally spaced and form a straight line along the grating vector. Note that the diffracted orders exist only when they are inside the circle of the medium refractive index (nair for reflected diffraction orders, nw for transmitted diffraction orders) in the relative direction cosine diagram since α2 + β2 ≤ n2 is always valid.
For the diffraction of light in various incident angles, the shape and size of the diffraction fields in the relative direction cosine diagram remain unchanged but monolithically shift in mλ0/d along the grating vector from the field of incidence, regardless of the refractive index change between the medium on the two sides of the grating, as shown in Fig. 2(c). In this way, the conical grating diffraction of the wide-angle incidence, representing the collimated light from an image, can be intuitively expressed by shifts in the relative direction cosine diagram. Moreover, the tedious computation of multiple wide-angle conical diffractions in the diffractive waveguide can be largely simplified by drawing a series of shifts of the field of incidence along the grating vector of each grating.
3. In-coupling grating design
Designing a diffractive waveguide begins with the in-coupling grating, for which two factors are influential; diffraction angle and efficiency. The diffraction angle is related to the vacuum wavelength (λ0) and the grating period (d). Meanwhile, the diffraction efficiency depends mostly on the incident angle, grating shape, and diffraction order. Incident angles are determined by the incident FOV. The grating shape requires exquisite manipulation, which is excluded from the scope of this work. With respect to diffraction orders, generally, ±1st-order diffraction contributes the most. Thus, higher diffraction orders have been omitted in this work, although it is possible to implement them in some gratings.
Suppose that light in-coupling occurs at the air–waveguide boundary, and the refractive indices are represented by nair and nw, respectively. The incident light can be regarded as a composition of collimated beams at various incident angles (θi, ϕi). Suppose that they are within the range named the incident FOV. The propagation of light through the waveguide relies on total internal reflection (TIR); otherwise, energy leaks quickly, and no perceivable light is left for out-coupling. This is the first restriction for in-coupling grating design. As mentioned in Section 2, in a relative direction cosine diagram, the region inside a circle of radius n indicates physically allowed light propagation in the medium of refractive index n. Thus, the TIR on the waveguide–air boundary is represented by an annulus region with an inner radius of ${n_{air}}$ and an outer radius of nw:
Usually, an incident FOV that meets the restriction above is regarded as the valid FOV. According to the relative direction cosine diagram, the valid FOV is limited by the size of the TIR region, or ultimately the difference between two medium refractive indices, nw–nair. Therefore, a high-refractive-index waveguide is required for large-FOV diffractive waveguides.
To explain the use of relative direction cosine design, we take the design of a transmission grating as an example. For clarity and simplicity, the following conditions are assumed:
- 1. The grating vector is on the x–y plane and toward the + x axis, which means θG = π/2 and ϕG = 0.
- 2. The incident FOV is circular and centered along $\hat{n}$ with a range of 0 ≤ θ ≤ θFOV and -π ≤ ϕ ≤ π, where θFOV is the largest polar incident angle and θFOV∈(0, π/2].
In the relative direction cosine diagram (Fig. 3), the incident FOV is a circle of radius nairsinθFOV, colored by red. The maximum possible incident FOV is the circle of radius nair, colored by light blue. After diffraction from the in-coupling grating (G1), the center of FOV is shifted from the coordinate origin by mλ0/d1, where d1 is the period of G1. Thus, the valid FOV, colored by green, is the annulus region shifted reversely from the TIR annulus region (colored by gray). Based on the relative direction cosine diagram, the valid incident FOV reaches its maximum when the center of the diffracted FOV is at the middle of TIR inner and outer boundaries. Therefore, the optimal period of G1 for a maximum FOV is calculated by
The corresponding maximum FOV, expressed by θFOV can be calculated by
Fig. 3. Relative direction cosine diagram. (a) Illustration of the valid FOV, maximum possible incident FOV, valid field of propagation, field of diffraction, maximum valid incident FOV, and valid incident FOV. (b) Field of incidence from rectangular and circular displays.
Notably, the incident FOV defined above is half the diagonal angle in the context of display projectors and different from the commonly called FOV of an AR device, which refers to the full diagonal angle. Equation (11) gives the minimum FOV that a waveguide can support for usual cases (no dividing FOV or multiplexing diffraction orders). This also gives room for optimizing the value of the grating period. In addition, since the microdisplay such as OLED and Micro-LED is rectangular rather than round, the field of incidence for the microdisplay is curved rectangular in the relative direction cosine diagram, as marked by the red line in Fig. 3 (b).
For a concrete explanation, Fig. 4 shows a numerical example of FOV calculation. Figure 4(a) shows the optimal period of the in-coupling grating (d1) as a function of waveguide refractive index (nw) according to Eq. (10) using wavelengths of 450, 520, and 635 nm representing blue, green, and red incident lights, respectively. The maximum FOV (θFOV) is also calculated as a function of nw by Eq. (11). For a commonly used high-refractive-index waveguide with nw = 1.7, θFOV reaches 20.5°, which enables the waveguide to carry a rectangular-shaped virtual image with a full diagonal FOV over 40°. Note that θFOV is limited by nw but is merely influenced by the material dispersion. The optimal period of the in-coupling grating depends highly on the wavelength. In the case of transferring an R/G/B image using a certain grating configuration in a single-layer waveguide, the period of the grating can only meet FOV maximization requirements of one wavelength, which severely limits the overall FOV that covers all colors. To solve this problem and improve the overall coupling efficiency, techniques including using achromatic diffractive elements or stacking multiple waveguides have been proposed [12,13,36–40].
Fig. 4. (a) Calculated optimal grating period (d1) and the corresponding θFOV as a function of nw when λ0 = 450, 520, and 635 nm. (b) Polar angles of the 1st-order diffraction (θ1) for various θi and d1 when ϕi = 0, nw = 1.7, and λ0 = 520 nm.
Figure 4 (b) shows the 1st-order diffracted polar angle (θ1) contour plot as a function of the incident polar angle (θi) and the in-coupling grating period (d1) when ϕi = 0, nw = 1.7, and λ0 = 520 nm. The blank area in the bottom-left corner depicts that the 1st-order diffraction does not exist under such conditions. The blank region in the top-right corner refers to invalid incidence, where θ1 is smaller than the critical angle and does not meet the TIR condition. To keep the incidence valid, a smaller grating period (d1) is required as θi increases. From the figure, for a valid incident FOV centered at θi = 0, the optimal period of the in-coupling grating is 385 nm, which agrees well with the optimal period of 385.2 nm calculated by Eq. (10).
4. Matching the turning and out-coupling gratings with the in-coupling grating
The out-coupled light must be at the same or opposite polar/azimuthal angles as the incident light to ensure that the image projected by the microdisplay can be well-restored in the viewer’s eye without distortion after the diffractive waveguide. This requires matching parameters among in-coupling and out-coupling gratings, as well as turning gratings. The periods, directions, and relative positions of gratings should be precisely designed. The matching can be achieved by analyzing the diffraction properties of gratings in a RDCS. The following sections demonstrate designs of various diffractive waveguide layouts and functionalities of gratings.
4.1 Grating matching in the diffractive waveguide with one-dimensional pupil expansion
In the simplest case, the diffractive waveguide is assembled by two gratings, one for in-coupling (G1) and the other for out-coupling (G2). The incident light is coupled from air into the waveguide through G1 and propagates toward G2. Then, after diffraction by G2, the light is out-coupled from the waveguide. Light propagation through the diffractive waveguide in a RDCS is illustrated in Fig. 5. The restoration of the FOV requires two converse movements of relative direction cosines caused by grating diffraction. Thus, G1 and G2 should have the same grating vectors; +1st order diffraction takes place on either grating and -1st order diffraction on the other, which gives d2 = d1. Meanwhile, such a configuration enables one-dimensional pupil expansion. On the out-coupling grating, despite diffraction, incident collimated beams are duplicated by reflection on the grating and then TIR on the waveguide–air boundary. Duplicated beams all carry the same information but enlarge the perceivable area for eyes (or exit pupil size, usually named eye box) along the propagation direction in the waveguide, although the intensity diminishes slowly.
Fig. 5. Relative direction cosine diagram of the diffractive waveguide with one-dimensional pupil expansion using in-coupling and out-coupling gratings. The inset shows the layout of the diffractive waveguide.
4.2 Grating matching in the diffractive waveguide with two-dimensional pupil expansion
By adding extra gratings, it is possible to fold the optical path in the waveguide and achieve two-dimensional pupil expansion. The extra gratings are called turning gratings or folding gratings. The easiest way is to add one turning grating (G2) on the optical path between the in-coupling grating (G1) and the out-coupling grating (G3) and set a grating direction for G2 different from that of G1.
Assuming that all three gratings and their grating vectors are on the x–y plane (θG = π/2), the azimuthal angles are ϕG1, ϕG2, and ϕG3, and the grating periods are d1, d2, and d3, respectively.
It is an important restriction for G2 that the diffracted light by G2 should be within the TIR region of the waveguide. Therefore, for simplicity, the central positions of FOVs diffracted by G1 and G2 are designed to share the same distance from the incident FOV in the relative cosine space, as shown in Fig. 6 (a). Consequently, d1 = d3, and the three positions of FOVs construct an isosceles triangle. After grating directions ϕG1, ϕG2, and ϕG3 are determined by the geometric requirements of the diffractive waveguide, d2 can be calculated using trigonometric functions. Figure 6 (b) and (c) illustrate two typical diffractive waveguide layouts in which G2 is rotated 45°/60° from the direction of G1 (ϕG2 - ϕG1 = 45°/60°). For 45° rotation, d2 = d1/√2. For 60° rotation, d2 = d1.
Fig. 6. (a) Relative direction cosine diagram of the diffractive waveguide with two-dimensional pupil expansion. (b)(c) Turning grating having rotation angles of 45° and 60°. The inset shows layouts of the diffractive waveguides (blue) and pupil expansion conditions (red dot circles).
4.3 Grating matching in the diffractive waveguide using multiple turning gratings or a two-dimensional grating
Using a relative-direction-cosines diagram is a convenient approach for more complicated diffractive waveguide designs with multiple turning gratings or two-dimensional gratings. Figure 7(a) shows a diffractive waveguide with four turning gratings to deliver both the 1st and -1st orders of the diffraction of G1 in two paths (G1–G2–G4–G6, G1–G3–G5–G6). In the relative direction cosine diagram shown in Fig. 7(b), the 1st-order diffraction of G1 is represented by a λ0/λ1 shift of the direction cosine variables along the +α axis. This portion of the light is then redirected by G2 and G4. In this case, G2 is rotated 45° clockwise with respect to G1. Accordingly, G4 is rotated 90° clockwise from G2, and thus d4 = d2 = d1/√2. The case of the -1st-order diffraction is similar. This design can improve the overall efficiency and illumination uniformity throughout the incident FOV as both 1st and -1st orders are adopted, and more parameters are available to balance the angular intensity distribution. In addition, by appropriate designs with multiple gratings, the incident FOV can be spatially divided into several parts and delivered by different grating paths to circumvent the valid FOV limited by the waveguide refractive index.
Fig. 7. Diffractive waveguide with (a) multiple turning gratings and (b) corresponding diffraction properties in the relative direction cosine diagram. Diffractive waveguide with (c) a two-dimensional grating and (d) the relative direction cosine diagram.
Figure 7(c) shows a diffractive waveguide with a two-dimensional grating implemented as a combination of turning gratings and an out-coupling grating. In the RDCS, the two-dimensional grating is treated as a superposition of two one-dimensional gratings (G2 and G2’). The FOV is delivered on two triangular paths (G1–G2’–G2, G1–G2–G2’), as shown in Fig. 7(d). Considering the TIR restriction, a simple but typical design involves G2 and G2’ having the same period as G1, and both grating vector directions of G2 and G2’ should be at an angle of 60° with respect to that of G1. As a result, the two-dimensional grating is similar to the square tiling of rhombi.
5. Results and discussion
To verify the viability of our design based on the relative direction cosine diagram, three-grating diffractive waveguides, as shown in Fig. 6 (b), were fabricated. The waveguides were made of glass (SCHOTT N-SF1) with a refractive index of 1.723 (@520nm) with ve = 29.39 and thickness of 0.35 mm. The gratings were made up of a photoresist (NTTAT #18210) with a refractive index of 1.735 using nanoimprinting (the imprint mold was fabricated by interference lithography using dry-etching technology). The periods of the gratings were d1 = 382, d2 = 270 and d3 = 382 nm (λ0 = 520 nm). The three gratings were all slanted gratings. The groove profiles, including groove depths, duty cycles, and slant angles (9 parameters in total), were optimized using RCWA and the non-dominated sorting genetic algorithm (NSGA-II) [41].
Figure 8 (a) illustrates the diffractive waveguide with a 520-nm light incident on the in-coupling grating (G1). The bright band from G1 shows the light propagation path along G2, proving that the light was effectively coupled into the waveguide by G1. Note that the observed propagation path was the light scattered by the glass and dust on the glass surface rather than the diffracted light from grating G2. The out-coupling grating (G3) was fully illuminated, indicating that the light beam was successfully expanded to the size of G3 and out-coupled by the diffraction of G3.
Fig. 8. Three-grating diffractive waveguide with d1 = 382, d2 = 270, and d3 = 382 nm with an incidence of (a) a laser beam (520 nm) and (b) a projection of the logo of XLOONG. The augmented reality image is observed through the diffractive waveguide.
Figure 8 (b) shows the AR image observed via G3 when a virtual image is incident toward G1 after collimation. The virtual image is generated by a digital light processing pico-projector, which has horizontal and vertical projection angles 35° (H) × 20° (V). For the demonstration of the optimization results based on 520-nm light incidence, only green light (central wavelength of 515 nm) is adopted to display the virtual image. We see that a logo of XLOONG is displayed with a real-word sense simultaneously. This proves the practicality of the diffractive waveguide and confirmed our method based on the relative direction cosine diagram. Note that the FOV of the virtual image in Fig. 8 (b) is limited by the pico-projector rather than the valid incident range of the waveguide.
The FOV measurement of the diffractive waveguide is shown in Fig. 9 (a). The three-grating waveguide is placed on a horizontal rotation stage and the in-coupling grating (G1) is at the center of the rotation stage. A 520-nm laser is fixed on a vertical rotation stage toward G1. The Cartesian coordinates describing incident angles are set where the origin is at the center of G1, and the z axis is along the normal direction of the gratings. The x axis is parallel with the intersecting line of the two rotation planes. Therefore, the polar angle (angle between the incident light and the z axis) is controlled by the vertical rotation stage. The azimuthal angle, expressed by the angle between the x axis and the grating vector of G1 in the x–y plane, is determined by the horizontal rotation stage. The out-coupled light from the diffractive waveguide is measured by an optical power meter (PM400 with slim sensor S130 from Thorlabs) 10 mm above the out-coupling grating (G3). An obvious increase in optical power is detected once the light is successfully in-coupled and out-coupled.
Fig. 9. FOV measurement of the diffractive waveguide. (a) Experimental setup. (b) Experimental results of the maximum polar angles that determine the FOV in various azimuthal angles.
Figure 9 (b) illustrates the theoretical and practical FOVs in a polar coordinate plot. The theoretical FOV, inscribed in a curved-quadrilateral-shaped valid field determined by both in-coupling and turning gratings (G1 and G2), is calculated as 43° and marked by the blue line in Fig. 9 (b). The practical valid field of the light incidence is obtained by measuring the maximum polar angle of the light incidence that effectively out-couples light from G3 when the azimuthal angle of the light incidence increases from 0 to 360°. As a result, the practical valid field (marked by the stars in Fig. 9 (b)) has a similar shape to the theoretical valid field but with a smaller range. Accordingly, the practical FOV is measured to be approximately 39° and slightly deviated from the center of view (∼ 0.5°). The decreased FOV and the deviation are caused by production errors, which make the periods of the gratings different from their optimal values. Note that the efficiency variance of different incident angles is not discussed here. For an incident angle near the right and left boundaries of the valid field, the diffraction efficiency of G1 and G2 is low. This causes the image edge to obscure when the projection angle reaches the maximum FOV of the diffractive waveguide. The FOV of a high-quality virtual image is thus slightly smaller than that measured using a laser beam.
Figure 10 shows a color image observed via the diffractive waveguide. The virtual image shows color deviation, where the green color is more apparent than the blue and red colors because the gratings are optimized for the light incidence with a wavelength of 520 nm, and the green light, which is nearest to this wavelength, shows the best display effect, as predicted by the theory. The color deviation can be prevented by adopting achromatic gratings or a metasurface, or more simply by stacking several diffractive waveguides optimized for different wavelengths [12,13,37–40].
6. Conclusions
In this paper, the relative direction cosine diagram was demonstrated as a convenient and effective tool for diffractive waveguide design. Firstly, the concept of the RDCS was proposed to describe the wide-angle conical diffraction behavior of gratings. Then, we demonstrated that the optimization of the gratings in a diffractive waveguide was greatly simplified using the relative direction cosine diagram that intuitively expresses the diffraction of the gratings by a series of shifts of relative direction cosines. Based on the relative direction cosine diagram, several equations were derived to precisely calculate the optimal periods of gratings and the corresponding FOV. Finally, three-grating diffractive waveguides were fabricated with optimal grating periods (382, 270, and 382 nm) for verification. The virtual image was observed through the diffractive waveguide without any obvious distortions, and the FOV was measured as 39° with 0.5° deviation from the center of view, which is in good agreement with the theory. Future work will demonstrate the application of relative direction cosine diagrams in optimizing diffractive waveguides with more complex layouts. Other design considerations such as efficiency optimization and color imaging will also be clarified.
Funding
Beijing Municipal Science and Technology Commission (Z191100004819005).
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.
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