1. Introduction

Optical metasurfaces have recently found a wide range of applications for their ability to shape wavefront with unprecedented flexibility. In contrast to conventional optics requiring multiple parts and bulky optical medium to modulate phase, amplitude and polarization of light wave, metasurfaces can implement these functions at a thickness comparable to or smaller than the wavelength of light through geometric phase [1,2] coupled modes [3,4] or dipolar resonances [5–7] of nanoscale light scatterers, thus opening the door to create new optical functionalities. In particular, metasurfaces have been widely studied to address the challenges of conventional optics for desired optical response at different wavelengths and show a considerable potential for practical applications to achieve broad-band efficient beam redirecting [8] and achromatic focusing [9].

Waveguide-based augmented realities (AR), as rapidly developing technologies to construct virtual images in front of human eyes, are currently bottlenecked by both weight and size of a stack of glass elements interfaced with a micro-projector for light coupling and diffracting at RGB wavelengths by employing conventional binary gratings on glass substrates with high refractive index. Large-scale penetration of AR will require lightweight optical components and low power consumption, ideally a metasurface structure on a single piece of eye glass for highly efficient RGB light coupling into the first-order diffraction mode across angular ranges of view. A special requirement in this application is that the beam deflection angles are high enough to facilitate total internal reflection in a glass substrate for image propagation (e.g., > 36° for glass with refractive index of 1.7). Therefore, the concept of metasurface holds the promise of replacing existing multiple glass waveguides with surface-relief gratings by form factors of a single-piece waveguide and its optical performances. For this purpose, planar effectively-blazed metasurfaces need to be designed to diffract incoming beams with a high angle into the first-order diffraction directions with nearly equal efficiencies for three primary colors. For conventional binary gratings, large deflecting angles usually contradicts with the deflecting efficiency: high refraction angles requires a steep linear phase profile, or small grating period p, while efficiency increases with the period-to-wavelength ratio p/λ [8,10]. For the conventional blazed grating, its diffraction efficiency significantly drops when the period is smaller than the incident light wavelength [11]. Although slanted grating was proposed to achieve a high diffraction efficiency across a field of view (FOV) [12–15], a full-color system using only one exit pupil expansion plate would be very difficult to implement because the diffraction angle depends very much on the wavelength. Metasurfaces offer a potential to provide a better solution to this coupling issue owing to resonances in different parts of its subwavelength structure at incidence of different wavelengths. However, it is still challenging to reach a design to allow uniform and efficient coupling of RGB primary colors for angles across FOV. Previously metasurface has reached a sufficient efficiency at the green light, but the diffraction angle is small because of a large period [2,16]. In addition, there have been some designs that can achieve a FOV of 40° and an over 80% average see-through transmittance using reflection volume holograms, but when it comes to the full-color display, it also needs to use two or three layers of waveguide plates for different wavelengths [17–19]. Additionally, metasurfaces or meta-gratings have been studied to replace conventional gratings with high coupling efficiencies in visible wavelengths [8], e.g., single in-coupler integrated with a polarization-multiplexed metagrating for stereoscopic images to different eyes [20], etc. Unlike metalenses [21], where eight or more subwavelength antennas were employed to modulate phase of incoming optical wave to form a wavefront similar to that through a Fresnel lens, metasurfaces for AR applications offers a trade-off between uniform diffraction efficiency and diffraction efficiencies at RGB wavelengths. In order to achieve a white balance across angles in FOV, nano-structures containing two or more field concentrations are often needed. These patterned beams can be generated using two or three nano-beams (with nano-sized rectangular cross-sections) for initial feasibility study.

Using metasurface that efficiently support large diffraction angles with a uniform efficiency at RGB wavelengths is unprecedented in the literature. A high and uniform efficiency across the angular range of FOV would be highly desirable for optical elements in near-eye display applications. In this work, we study the feasibility to diffract beam with a uniform efficiency at RGB wavelengths using a single layer of metasurface. We focus on the optimization process of nano-beam parameters of metasurfaces for a flat coupling efficiency at RGB primary colors throughout angular range of FOV. A uniform coupling efficiency as high as 30-40% for RGB wavelengths across FOV is obtained in full-vector simulation using FDTD for two types of metasurfaces. Based on the combination of wavefront deflection and near-field intensity modulation created by localized field resonance, metasurfaces of both double and triple nano-beams are investigated in terms of geometric parameters in an effort. Theoretical analysis based on field strength and phase distribution are performed to illustrate the effectiveness of the proposed metasurface structure, and an optimal parameter setting is reached using detailed contour maps of structure parameters.

2. Structure and design method

Figure 1(a) shows a layout of our metasurface based waveguide coupler for AR applications, where only one-dimensional pupil expansion with an extended output image area is displayed for simplicity. An in-coupling metasurface grating couples images consisting of R, G, and B primary colors from a micro-projector into a glass waveguide, wherein the three-color optical beams propagate horizontally via a zig-zag path to the out-coupling metasurface grating in front of the viewer’s eye. Both gratings are fabricated on the same side of the waveguide and working in the transmission mode. Meanwhile, the real-world scene, transmitting through first the glass substrate and then the out-coupling grating, is received by the human eye. The metasurface structure is designed to efficiently diffract energy of the incident light into the first diffraction order. The first-order diffraction angles differ for RGB beams, but all are required to be greater than the critical angle for total internal reflection at glass/air interfaces, enabling image transfer between the two gratings. The out-coupling grating, normally featuring a relatively large area for a comfortable eyebox (a space where the user’s pupil can be accommodated and view the entire virtual image over the FOV), has a unit cell structure identical to that of the in-coupling grating to recover the images, while its efficiency is usually reduced and modulated to achieve brightness uniformity of images. To have good fidelity of the recovered images, the first-order diffraction efficiencies of the gratings have to be balanced among the three colors. In addition, an adequate FOV is required for AR devices, which needs to be addressed along with color balance.

 

Fig. 1. (a) Optical design layout of our proposed AR glass for head mounted display. Three-dimensional schematic diagrams for our (b) double nano-beam and (c) triple nano-beam gratings. The geometrical parameters, the light incident angle as well as the first-order diffraction angle are defined in the two-dimensional cross-sectional schematic diagrams of the (d) double nano-beam and (e) triple nano-beam gratings, respectively.

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For conventional gratings with only one beam in each period (unit cell), it is nearly not possible to realize simultaneously both high coupling efficiencies for all RGB colors and a wide FOV. Much research has shown that diffraction efficiency of conventional gratings depends strongly on wavelength, beam/groove depth, and period [22], therefore a trade-off between diffraction efficiency of RGB colors and range of FOV is hard to obtain in a traditional dielectric grating. At present most waveguide-based AR glasses utilize configuration containing two-piece or three-piece gratings and each waveguide has its own designated gratings for specific working wavelength(s), because the grating dispersion spreads the angles more than what the waveguide can support and causes leakage at other wavelengths [23]. In this work, we propose a design of metasurface structures with either two or three nano-beams in each period for a simultaneous coupling of RGB light, as shown in Figs. 1(b) and 1(c), respectively. For these structures, it is seen from Figs. 1(d) and 1(e) that in each period (p) the cross-sections of the beams are all rectangular with the same thickness (t) but different widths (w₁, w₂, w₃). Gap sizes (g) between adjacent beams in each period are set to be the same for simplicity. Compared with conventional gratings, both the double and triple nano-beam metasurfaces have more subwavelength geometrical components, allowing to be optimized to modulate RGB optical waves separately to meet the aforementioned requirements.

As the nano-beam lengths are much longer than their widths as well as the grating period, a two-dimensional full-vector numerical simulation model was built based on a finite-difference time domain (FDTD) method using the commercial software Lumerical FDTD Solutions. As shown in Figs. 1(d) and 1(e), an s-polarized plane wave is incident onto the gratings at an angle of θ, and part of the transmitted energy is diffracted into the first diffraction order with an angle of α. The R, G, and B light wavelengths are λ₃ = 620 nm, λ₂ = 530 nm and λ₁ = 460 nm, respectively, which are commonly used in the field of LED illuminated micro-projector (e.g. RGB SMT modules from OSRAM, LE RTDUW S2W or LE ATB S2WWL). In simulation the first-order diffraction efficiency was obtained by applying grating function by projecting the near-field optical field to the far field. The efficiency evaluates how much energy of the input light is coupled to the first diffraction order.

Before conducting the numerical simulation, we first estimate the range of grating period, p, and FOV using following grating equation for the first-order transmitted diffraction [24],

Meanwhile, Eq. (2) holds across FOV, denoted as 2θ_FOV and ranging from -θ_FOV to θ_FOV. Therefore, Eq. (2) can be further expressed as:

To ensure that the value of p exists in this inequality, it follows that θ_FOV is a function of RB wavelengths and refractive index of waveguide material:

As indicated in Fig. 2(a), FOV increases approximately linearly as n_s increases. The large FOV will reduce the period range according to Eq. (4), which may also degrade the first-order diffraction efficiency. As shown in Fig. 2(b), the glass substrate with n_s = 1.85 corresponds to the critical angle of total internal reflection of α_c = 32.7°, as shown in Fig. 2(b). In order to enable the first-order diffracted RGB lights to fulfill the total internal reflection condition, FOV cannot be larger than 2θ_FOV = 24.7°, as shown in Fig. 2(a). Although FOV from 40 to 50° can be realized [25], its diffraction efficiency decreases rapidly upon illumination at oblique incidence angles. Moreover, an even large FOV (over 100°) was also achievable [26], but the diffraction angle is not large enough to meet the condition of total internal reflection. These nano-structures are not feasible for diffracting beam with a uniform efficiency among all RGB wavelengths. We will show in the following optimization process that this parameter selection would maintain high diffraction efficiencies for RGB primary colors. In this case, we could solve for p_max= 380.8 nm and p_min= 377.6 nm by Eq. (4). For glass substrate with n_s = 2.0, as shown in Fig. 2(a), FOV will be extended up to 32.3°.

 

Fig. 2. (a) Maximum FOV as a function of the refractive index of the waveguide accommodating RGB colors, n_s. When n_s = 1.85, the horizontal FOV is 24.7°, denoted as a green triangle. (b) The first-order diffraction angle as a function of the incident angle for all of RGB lights when the grating period p = 380 nm. When n_s = 1.85, the critical angle of total reflection angle α_c = 32.7°, denoted as a black dashed line.

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We chose p = 380 nm as the grating period, the first-order diffraction angle was calculated according to Eq. (1) and plotted in Fig. 2(b) for all light wavelengths across FOV. It is seen that all diffraction angles, α, are greater than α_c, which allow a horizontal FOV of about 24.7°, as well as lossless propagation of RGB lights via total internal reflection. As mentioned before, the out-coupling grating has unit cell identical to that of the in-coupling grating, whose first-order diffracted light becomes the incident light to the out-coupling grating with incident angle larger than α_c. According to the classical grating equation, there will be no zero-order diffraction exiting from the out-coupling grating. Theoretically the human eye only receives the strong first-order diffracted light without any interferences from other diffraction orders.

3. Simulation results and discussions

3.1 Uniform efficiencies for RGB primary colors

Based on the above analysis, we set glass substrate n_s = 1.85 and p = 380 nm for gratings considered in this work. Numerical simulations were performed to optimize the remaining parameters of the metasurface gratings to achieve high and uniform first-order diffraction efficiencies among RGB colors in the entire FOV ranging from -12.4° to 12.4°. Genetic algorithm [27] is usually applied in multiple-parameter optimization. However, it is very time-consuming. Here we performed a simplified optimization process to obtain the optimal geometrical parameters, including nano-beam widths, thickness, and gap sizes. We first set all nano-beam widths to 1/4 of the period, i.e., w₁ = w₂ = w₃ = 95 nm, and then scanned the nano-beam thickness, t, and adjacent beam gap, g, simultaneously to calculate RGB first-order diffraction efficiencies at three representative incident angles, i.e., -12°, 0° and 12°. The combination of t and g values were chosen for the highest and the most uniform RGB efficiencies at the three incident angles. Next, we set t and g to the chosen values, and used the same methods to optimize the widths of different nano-beams. Finally, all the geometrical parameters were set to the chosen values and the grating performance in the entire FOV were verified at RGB wavelengths. Though this optimization method might not be thorough, we achieved what were expected, as demonstrated in the following. For comparison, conventional single-beam gratings were also simulated and optimized. Since it is nearly impossible to achieve uniform efficiencies among all the RGB colors, two stacked single-beam gratings are employed for testing the efficiency balance, each piece for two of three wavelengths, namely RG and GB, respectively. This is routine for designing AR glass using two-piece configuration. Figure 3 shows the first-order diffraction efficiencies at RGB three operating wavelengths of the four optimized gratings for the incident angle of light in the FOV range. Dielectric material of a-Si is used for meta-structure in this study for the convenience of follow-up experimental verification. The structural parameters are w₁ = 120 nm, t = 55 nm for the conventional GB single-beam grating (Fig. 3(a)), w₁ = 116 nm, t = 75 nm for the conventional RG single-beam grating (Fig. 3(b)), w₁ = 15 nm, w₂ = 85 nm, t = 65 nm, g = 60 nm for the double nano-beam grating (Fig. 3(c)) and w₁ = 20 nm, w₂ = 20 nm, w₃ = 64 nm, t = 95 nm, g = 54 nm for the triple nano-beam grating (Fig. 3(d)), respectively. These parameters are used in this work unless otherwise specified.

 

Fig. 3. The first-order diffraction efficiencies at RGB three operating wavelengths of four optimized gratings, i.e., the conventional (a) GB and (b) RG single-beam gratings, as well as (c) the double-beam grating and (d) the three-beam grating, when the incident angle varies in the entire FOV from -12.4° to 12.4°. The structural parameters are w₁ = 120 nm, t = 55 nm for the conventional GB single-beam grating, w₁ = 116 nm, t = 75 nm for the conventional RG single-beam grating, w₁ = 15 nm, w₂ = 85 nm, t = 65 nm, g = 60 nm for the double nano-beam grating, w₁ = 20 nm, w₂ = 20 nm, w₃ = 64 nm, t = 95 nm, g = 54 nm for the triple nano-beam grating, respectively. The unit cell of the four gratings with n_s= 1.85 and p = 380 nm are schematically plotted in the inset of each panel.

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The first-order diffraction cannot be balanced with high efficiencies among RGB three colors for conventional single-beam gratings. As shown in Fig. 3(a), for the optimized GB grating, the first-order diffraction efficiencies at λ₂ = 530 nm and λ₁ = 460 nm are almost the same, which are much higher than that at λ₃= 620 nm, but lower than 0.3 over the whole FOV range (Fig. 3(a)). It is seen in Fig. 3(b) that a little bit higher and almost equal first-order diffraction efficiencies (above 0.3) are obtained at λ₃ = 620 nm and λ₂ = 530 nm for the optimal RG grating with the incident angle ranging from -12.4° to 6°. Beyond 6°, the efficiency of the red light decreases more quickly than that of the green light (Fig. 3(b)). The efficiency of the blue light stays below 0.2 in the whole FOV, deviating much from those of the red and green lights. Therefore, with these GB and RG single-beam gratings for light in-coupling, at least two pieces of AR glass are necessary for uniform image delivery. However, the whole device becomes very complicated and bulky.

In contrast, by using more nano-beams in a unit cell, the problems with the conventional single-beam gratings can be well addressed. As shown in Fig. 3(c), the double nano-beam grating can diffract the RGB light energy into the first diffraction order with the efficiencies over 0.3 and the efficiency deviations below 0.1 across entire FOV. In this case, only a single piece of AR glass substrate is adequate for high-efficiency image transmission with uniform brightness, which greatly enhances the viewer’s comfort. Further increasing the beam number to three in a unit cell will also allow us to achieve high and uniform first-order diffraction among the three primary colors, as shown in Fig. 3(d). However, relatively lower efficiencies than 0.3 (0.3-0.25 and 0.3-0.28, respectively) are observed for the red and blue lights for incident angles greater than about 9°. Therefore, the overall performance of the double nano-beam grating is superior to that of the three nano-beam grating.

It is known that a-Si is intrinsically absorptive in the visible regime, especially for the blue light. This may cause undesirable absorption in the grating, negatively affecting the first-order diffraction efficiency. To quantitatively investigate these effects, we calculated the absorptance as a function of incident angle at RGB wavelengths for the four gratings, as shown in Fig. 4. The absorptance is obtained through subtraction of reflectance and transmittance from one. In Fig. 4, all gratings show greater absorption of blue light than those of green and red lights. Even at λ₁ = 460 nm, the absorption is still lower than 0.5. This is attributed to the thin designs of the gratings with thicknesses less than 100 nm. For the triple nano-beam grating, the RG absorption are close to each other, with a slightly high absorption at λ₃ = 620 nm when the incident angle is greater than 9° (Fig. 4(d)). This is likely for the reason that the first-order diffraction efficiency is lower than 0.3 in that angle range (Fig. 3(d)). More importantly, such a wavelength-dependent absorption is largely due to the resonance-induced light-matter interaction and is able to positively modulate the diffraction efficiency among RGB waves so as to produce a uniform response (to be explained in detail below).

 

Fig. 4. Absorptance at RGB three operating wavelengths of four optimized gratings, i.e., the conventional (a) GB and (b) RG single-beam gratings, as well as (c) the double and (d) triple nano-beam gratings when the incident angle varies across the entire FOV from -12.4° to 12.4°. The structural parameters are w₁ = 120 nm, t = 55 nm for the conventional GB single-beam grating, w₁ = 116 nm, t = 75 nm for the conventional RG single-beam grating, w₁ = 15 nm, w₂ = 85 nm, t = 65 nm, g = 60 nm for the double nano-beam grating, w₁ = 20 nm, w₂ = 20 nm, w₃ = 64 nm, t = 95 nm, g = 54 nm for the triple nano-beam grating, respectively. The unit cell of the gratings with n_s= 1.85 and p = 380 nm are schematically plotted in the inset of each panel.

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3.2 Analysis of underlying physics

In order to understand the above interesting diffraction behaviors, we investigated into the near-field using electric field intensity, light energy flux described by Poynting vector and phase distributions of the four gratings. Under normal incidence these distributions are displayed in Figs. 5 and 6, respectively. As shown in Fig. 5(a), for the GB single-beam grating, there is a strong electric field localized at the bottom of the rectangular nano-beam, which can be viewed as a resonator resonating with the blue light. Interacting with the resonator, the incident light is scattered and coupled to the first diffraction order. Meanwhile, strong resonance also lead to local absorption in a-Si (Fig. 4(a)), which inevitably degrades the diffraction efficiency to lower than 0.3 (Fig. 3(a)), despite the strong scattering. As the wavelength of light increases, the resonance becomes weak due to mode mismatch. Consequently, less transmitted energy is coupled to the first diffraction order and more light remains propagating downward vertically (Figs. 5(b) and 5(c)). Without strong absorption at λ₃ = 620 nm (Fig. 4(a)), the low diffraction efficiency is mainly induced by the weak scattering of light. In comparison, as the nano-beam size becomes large (with almost the same beam width but increased thickness) in the case of the RG single-beam grating, strong resonances occur at λ₂= 530 nm and λ₃ = 620 nm as shown in Figs. 5(e) and 5(f), respectively. Though absorption increases slightly (Fig. 4(b)), there is still a big portion of energy coupled to the first diffraction order (Fig. 3(b)). While at λ₁ = 460 nm, a higher-order resonance is excited, where scattering effect is much weaker. The light-matter interaction is not so strong, but the high intrinsic absorption of a-Si still make the RG grating as absorptive as the GB grating (Figs. 4(a) and 4(b)). The high absorption and weak scattering result in much lower first-order diffraction efficiency than those at red and green wavelengths (Fig. 3(b)). From these analyses, it is known that the first-order diffraction efficiency is modulated by optical resonances. This is a two-side effect: the diffraction efficiency can be either improved through resonance-induced scattering, or degraded by resonance-enhanced absorption. The optical resonances strongly depend on the beam size and the illuminating wavelength. Due to a large wavelength span of 160 nm between red and blue illumination light, a single rectangular beam in a period can hardly support strong optical resonances as well as uniform diffraction efficiency at all of the RGB wavelengths.

 

Fig. 5. Electric field intensity (|E|²) distributions in the xz plane of (a-c) the conventional GB single-beam grating, (d-f) the conventional RG single-beam grating, (g-i) the double nano-beam grating, and (j)-(l) the triple nano-beam grating, respectively. The lights with wavelengths of (a, d, g, j) λ₁ = 460 nm, (b, e, h, k) λ₂ = 530 nm, and (c, f, i, l) λ₃ = 620 nm are normally incident on the gratings. The white arrows over the electric field distributions represent time-averaged Poynting vectors. The black dashed lines in each panel outline the unit cell structure of the grating.

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Fig. 6. (a) Phase distributions in the xz plane of (a-c) the conventional GB single-beam grating, (d-f) the conventional RG single-beam grating, (g-i) the double nano-beam grating, and (j)-(l) the triple nano-beam grating, respectively. The lights with wavelengths of (a, d, g, j) λ₁ = 460, (b, e, h, k) λ₂ = 530, and (c, f, i, l) λ₃ = 620 nm are normally incident on the gratings. The black dashed lines in each panel outline the unit cell structure of the grating.

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When two rectangular nano-beams with different sizes are introduced into a unit cell of the grating, for the short wavelength at λ₁ = 460 nm, the narrow beam would serve as a specific fundamental-mode resonator and the wide beam as a resonator for the rest of modes (Fig. 5(g)); while at the long wavelengths of λ₂ = 530 nm and λ₃ = 620 nm, the narrow beam can hardly accommodate the modes and the resonances are mainly excited in the wide beam (Figs. 5(h) and 5(i)). In contrast with the bilateral symmetric phase profiles of the conventional GB and RG single-beam grating (Figs. 6(a)–6(f)), the phase profile at each wavelength becomes asymmetric and appears to rotate clockwise by an angle determined by ratio of two nano-beam sizes (Figs. 6(g)–6(i)). As shown in Figs. 5(g)–5(i), the Poynting vectors that denotes light energy flux for the three colors are all pointing to the lower right corners even under normal illumination, leading to efficient coupling of the incident energy into the first diffraction order (Fig. 3(c)). On the other hand, as incident wavelength increases, the resonance of light coupled to nano-beams becomes weak, meanwhile the absorption loss of nano-beam also decreases (Fig. 4(c)). The balance between the scattering and the absorption loss leads to almost equal diffraction efficiencies among the three primary colors (Fig. 3(c)).

The triple nano-beam grating behaves similarly. The optical field is mainly localized in two of the three beams depending on matched resonance between wavelength and size of nano-beams, i.e., in the left two narrow beams for the blue light (Fig. 5(j)), in the left narrow and right wide beams for the green light (Fig. 5(k)), and in the middle narrow and right wide beams for the red light (Fig. 5(l)), respectively. The apparently asymmetric optical field distribution leads to a rotated phase profile or a deflected wavefront (Figs. 6(j)–6(l)) and thus a distribution of deflected Poynting vectors (Figs. 5(j)–5(l)). Hence, the diffraction efficiency is enhanced for all the RGB colors under normal incidence compared with those of the conventional single-beam gratings (Fig. 3(d)). The high absorption at λ₁ = 460 nm (Figs. 4(c) and Figs. 4(d)) leads to a nearly equal low efficiency of the first-order diffraction for both the double and triple nano-beam gratings (Figs. 3(c) and Figs. 3(d)). Among the four gratings, the red light absorption is the strongest for the triple nano-beam grating (Fig. 4(d)) due to the large light-matter overlapping area (Fig. 5(l)). Its diffraction efficiency is just above 0.3 (Fig. 3(d)), lower than that of the double nano-beam grating (Fig. 3(c)).

Regarding cases of oblique incidence, simulation results both for the double nano-beam and triple nano-beam gratings are studied. We display those cases that the illumination angles are set at the boundaries of the FOV, i.e., -12° and 12°. For the double nano-beam grating, when the incident angle changes from 0° to -12°, despite the slight clockwise rotation of the optical fields (Figs. 7(a)–7(c)), the resonance positions and intensities (Figs. 7(a)–7(c)), as well as the phase profiles (Figs. 8(a)–8(c)) keep almost unvaried for all the RGB colors. Therefore, the wavefronts and the Poynting vectors are almost the same as that under the normal incidence at each color. On the other hand, when the incident angle is changed to 12°, the field intensities are enhanced (Figs. 7(d)–7(f)) and the phase profile rotate counterclockwise (Figs. 8(d)–8(f)). The enhanced resonance causes increased absorption loss at λ₁ = 460 nm (Fig. 4(c)), where the intrinsic absorption of the a-Si reaches the highest. In this case, the coupled energy into the first diffraction order decreases slightly. For the case at λ₃ = 620 nm, the resonance in the narrow beam is enhanced so greatly (Fig. 7(f)) that the absorption is increased (Fig. 4(c)) and the phase is apparently changed (Fig. 8(f)). Since some portion of the light energy is lost by absorption and some goes back to the air (Fig. 7(f)), the first-order diffraction efficiency is degraded. The triple nano-beam grating in the oblique incident cases has a similar trend in terms of the optical field and phase profiles, as shown in Figs. 7(g)–7(l) and Figs. 8(g)–8(l), respectively. The diffraction efficiency at λ₃ = 620 nm under 12° incidence is obviously lower than that of the double nano-beam grating (Figs. 3(c) and 3(d)), which is mainly attributed to the high resonance-induced absorptance (Fig. 4(d)).

 

Fig. 7. Electric field intensity (|E|²) distributions in the xz plane of (a-f) the double nano-beam grating, and (g)-(l) the triple nano-beam grating, respectively. The lights with wavelengths of (a, d, g, j) λ₁ = 460, (b, e, h, k) λ₂ = 530, and (c, f, i, l) λ₃ = 620 nm are obliquely incident on the gratings with angles of (a-c, g-i) -12° and (d-f, j-l) 12°. The white arrows over the electric field distributions represent time-averaged Poynting vectors. The black dashed lines in each panel outline the unit cell structure of the grating.

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Fig. 8. Phase distributions in the xz plane of (a-f) the double nano-beam grating, and (g)-(l) the triple nano-beam grating, respectively. The lights with wavelengths of (a, d, g, j) λ₁ = 460, (b, e, h, k) λ₂ = 530, and (c, f, i, l) λ₃ = 620 nm are obliquely incident on the gratings with angles of (a-c, g-i) -12° and (d-f, j-l) 12°. The black dashed lines in each panel outline the unit cell structure of the grating.

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In general, by introducing multiple rectangular nano-beams into a grating unit cell, high and uniform first-order diffraction efficiency at all the RGB wavelengths in the entire FOV can be achieved through reasonable manipulation of both scattering and absorption induced by optical resonances. The resonance-induced scattering plays an important role in efficient coupling of light into the first diffraction order, while the resonance-enhanced absorption adjusts the efficiencies among RGB three colors. From this point of view, an absorptive grating may be helpful in generating uniform RGB diffraction efficiencies.

3.3 Structural effects

For further explanation, the structural effects of the representative double nano-beam grating were investigated for the first-order diffraction efficiency. As mentioned above, the nano-beams in the unit cell can be seen as resonators, where the optical resonances greatly depend on the geometrical parameters of the nano-beams. Therefore, we first presented in Fig. 9 the first-order diffraction efficiencies as functions of the widths of the two nano-beams under RGB illuminations with incident angles of θ = -12°, 0°, and 12°, respectively. From Fig. 9, it is seen that for the nine cases under different illumination conditions, the highest efficiencies all exist at the points where w₂ is larger than w₁. This means that the asymmetrical beam widths are beneficial for deflection of wavefront and thus the light coupling into the first diffraction order. Such energy transfer is less sensitive to the narrow beam width, w₁, than to the wide beam width, w₂, which can be clearly seen from the horizontally extended contour curves in Fig. 9. For a given narrow beam with a certain width, e.g., w₁ = 15 nm, the diffraction efficiency first increases and then decreases with the increasing wide beam width under RGB illuminations at all the incident angles. The value of w₂ corresponding to the maximal efficiency increases with the increasing light wavelength for a given incident angle. This is a typical feature of optical resonance, which requires a larger space to accommodate longer wavelength photons. If w₂ is too small, e.g., w₂ = 30 nm, the resonance cannot be well excited in both beams in the unit cell; while if w₂ is too large, e.g., w₂ = 85 nm, higher-order-resonances can be excited in the wider beam with relatively weak scattering of light. Consequently, the phase or wavefront cannot be deflected significantly (not shown here) and thus the coupling ratio of the incident light to the first diffraction order is low. The wide beam is bound to increase the absorption loss, also negatively affecting the light diffraction. The same observation holds well for both blue and green light illumination cases (Figs. 9(a)–9(f)). For the red light with the longest wavelength of λ₃ = 620 nm, the first-order diffraction angle, which increases nonlinearly with the increasing incident angle, is much greater than those of the blue and green lights, especially at the boundary of the FOV, as shown in Fig. 2(b). It needs much wider beams or stronger resonance to gain enough wavefront deflection for efficient energy transfer. Therefore, w₂ = 85 nm is not large enough to enable a maximal diffraction efficiency for the oblique incidence with θ = 12°, as shown in Fig. 9(i).

 

Fig. 9. The first-order diffraction efficiency as functions of the widths of two nanobeams, i.e., w₁ and w₂, of the double nano-beam grating when the lights with wavelengths of (a, d, g) λ₁ = 460, (b, e, h) λ₂ = 530, and (c, f, i) λ₃ = 620 nm are incident from the top at the angles of θ = -12° (a-c), 0° (d-f) and 12° (g-i), respectively. The black star in each panel indicates the chosen values of w₁ and w₂ for the optimal design. Here the beam thickness and the adjacent beam spacing are set to be t = 65 nm and g = 60 nm, respectively.

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From the point of view of practical application in AR glasses, efficiencies of the double nano-beam grating with w₁ = 15 nm and w₂ = 85 nm (marked as black stars in Fig. 9) are optimal at all the wavelengths and incident angles. At this point, the resonance-induced scattering and absorption are well traded off in the nine cases. The first-order diffraction efficiencies are almost equal with a value ranging from 0.3-0.4, which is extremely important for uniform image delivery and comfortable view. In each panel of Fig. 9, there is a very large area around the star where the diffraction efficiency varies within 20%, indicating a high degree of tolerance to the nano-beam widths. Therefore, we chose w₁ = 15 nm and w₂ = 85 nm for the design of the double-beam configuration, as demonstrated in Sections 3.1 and 3.2.

The beam thickness, t, is another critical geometrical parameter affecting the optical resonances. Similar to the trend of the first-order diffraction efficiency with the nano-beam widths shown in Fig. 9, the efficiency first increases and then decreases with the increasing beam thickness for all the nine cases under different illumination conditions, as shown in Fig. 10(a). For a given incident angle, as the light wavelength increases, the maximal efficiency increases and the thickness associated with the maxima also increases. Such wavelength-dependent characteristics show the existence of optical resonances in the nano-beams, allowing the energy transfer from the incident light to the first diffraction order. The spread of efficiency for a given wavelength within FOV is about 5% for blue and green light, and 8-12% for red light. Particularly, the efficiency curves converge at thickness t = 60-65 nm. This is beneficial for AR glass to achieve a RGB uniform efficiency across FOV. It is also seen in Fig. 10(a) that around t = 65 nm, the blue diffraction efficiencies reach their maxima under the three incident angles, and the green and red efficiencies do not deviate much. Nano-beams with high thickness will introduce loss due to the intrinsic absorption of dielectric material (a-Si in this case) and will also increase difficulty in fabrication. Therefore, it is reasonable to choose t = 65 nm as the optimal beam thickness, as stated in Sections 3.1 and 3.2.

 

Fig. 10. The first-order diffraction efficiency as functions of (a) the nano-beam thickness, t, with w₁ = 15 nm, w₂ = 85 nm, and g = 60 nm; and (b) the adjacent nano-beam spacing, g, with w₁ = 15 nm, w₂= 85 nm, and t = 65 nm, when the lights with wavelengths of λ₁ = 460 (blue curves), λ₂ = 530 (green curves), and λ₃ = 620 nm (red curves) are incident from the top with the angles of θ = -12° (dotted curves), 0° (solid curves) and 12° (dashed curves), respectively.

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The interactions between optical resonances in individual beams in the unit cell are mainly reflected by the adjacent beam spacing, g, as defined in Fig. 1(d). As shown in Fig. 10(b), when g varies from 20 to 60 nm, the first-order diffraction keeps in a narrow range for RGB colors with three different incident angles. While for g > 60 nm, efficiencies decline at a slow slope. This means that photons resonate independently in the two beams with nearly no interactions between them, which allows a wide range of g values to be applied to the grating design. In this case, fabrication tolerance becomes much acceptable. Here we chose g = 60 nm for our design.

3.4 Imaging of real-world scene

For AR applications, it is of great necessity to ensure image quality of the real-world scene. As shown in Fig. 1(a), when the real-world light transmits through the glass substrate and hits the out-coupling grating, it can also be diffracted and the zeroth-order diffracted light will be received by the human eye. Here we calculated efficiencies of the zeroth-order diffraction of the real-world RGB lights, assuming that the out-coupling grating is set to the conventional GB and RG single-beam gratings as well as the double and triple nano-beam gratings with optimized geometrical parameters mentioned in Section 3.1. The results are summarized in Table 1. It is seen that the outside RGB lights are dimmed to different degrees. Interestingly, only the double nano-beam grating has quite uniform responses (0.42-0.44) to RGB colors with deviation of only 0.02, while the other three gratings exhibit much greater non-uniformities from 0.38 to 0.78. This is clearly another advantage of our double nano-beam grating compared with others. It is noted that these zeroth-order efficiencies are only for the purpose of mutual comparison using the nano-structures mainly designed for in-coupling. In practice, out-coupling gratings normally reduces grating height (thus low overall first-order out-coupling efficiency) for image uniformity across eyebox, therefore the actual transmission would be much higher than those listed in Table 1.

Table 1. Calculated zeroth-order diffraction efficiencies of real-world RGB lights for different gratings all with optimized geometrical parameters discussed in Section 3.1.

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4. Conclusion

In conclusion, a metasurface grating with two and three nano-beams in each period is proposed in this study to achieve a uniform efficiency at RGB wavelengths. In simulation a nearly flat efficiency as high as 30-40% is obtained for the first diffraction order across entire FOV of 24.7°. For glass substrate with n_s = 2.0, FOV is expected to further extend to 32.3°. Such optical properties may deliver uniform images with only a single-layered metasurface grating in a waveguide-based AR glasses, enabling a light-weight and energy-efficient AR system. Systematic full-vector numerical simulations are performed and show that the strong resonances inside multiple nano-beams with different sizes are the main reason for the deflection of the phase or wavefront as well as the Poynting vector, leading to efficient coupling of the incident light to the first diffraction order. The resonances also manipulate the light absorption among the RGB colors, which makes the uniform first-order diffraction efficiency feasible with a configuration of double nano-beam grating. In order to provide more profound understanding of the metasurface physics, analysis of conical diffraction will be conducted in our future work. For RGB lights, the uniform efficiency of our grating is sufficiently high, though it is still lower than the efficiency of the grating operating at a single wavelength (e.g., the slanted grating at green light [12]). It is highly feasible to further improve the RGB uniform efficiency by replacing the absorptive a-Si with less absorptive and high-refractive-index materials. We believe that couplers with uniform RGB efficiencies will become more competitive than that working at a single wavelength in the near future because it enables simple, light-weight and comfortable AR devices.