1. Introduction

Optical metasurfaces, composed of subwavelength structures, offer versatile wavefront shaping capabilities through various mechanisms such as geometrical phase [1,2], coupled modes [3,4], or resonances [5–7] of nanoscale light scatters. Unlike traditional optics, they eliminate the need of multiple components and bulky mediums, enabling compact and lightweight designs. Therefore, optical metasurfaces have attracted significant interest for practical applications, addressing challenges in conventional optics such as broadband beam redirecting [8] and achromatic focusing [9]. Among these applications, waveguide-based augmented reality (AR) glasses have gained prominence due to their high optical transparency, design flexibility, and cost-effectiveness. In waveguide-based AR glasses, the in-coupling grating on the glass substrate plays a crucial role in efficiently coupling RGB primary colors from a micro-projector into the high-refractive-index glass waveguide. The primary requirement for such gratings is to achieve uniform and efficient coupling of RGB primary colors into the first diffraction order across a wide field of view (FOV). Additionally, to ensure lossless propagation within the waveguide, great deflection angles are necessary for each RGB color to enable total internal reflection. Achieving these objectives would enable lightweight and high-quality AR glasses with a single piece instead of multiple stacked layers. However, achieving uniform coupling efficiencies for RGB primary colors throughout the entire FOV poses a significant challenge.

Conventional gratings with one beam per period exhibit diffraction efficiency dependent on wavelength, beam height, and period [8,10,11]. Achieving balanced diffraction efficiencies for RGB colors across the FOV is extremely challenging, resulting in weakened intensity and a subpar visual experience for one or two colors. Consequently, waveguide-based AR glasses often require multiple waveguide plates, each operating at specific wavelengths [12]. However, this approach leads to thick and heavy AR glasses that are inconvenient to wear. For a conventional binary grating to obtain a high deflection angle, a small grating period, p, is preferred, but is unfavorable for a high efficiency, which is proportional to the period-to-wavelength ratio, p/λ [8,11]. When p/λ < 1, the diffraction efficiency dramatically decreases for a conventional blazed grating [13]. Slanted gratings have been proposed to achieve high diffraction efficiency across the FOV [14–17], they still face challenges with nonuniform coupling. Metasurfaces have emerged as potential solutions due to their ability to excite optical resonances at different wavelengths. However, the metasurface grating achieving a sufficient efficiency at 532 nm had a small diffraction angle because of a large period [2,18]. Some designs can achieve a FOV of 40^o and an average see-through transmittance over 80% through reflection volume holograms, but they do not enable full-color display, requiring multiple layers of waveguide for different wavelengths [19–21]. To address these challenges, a metasurface design was proposed theoretically [22]. However, the underlying physical mechanism needs to be investigated further. The method based on unit structure and geometric parameter optimization is common for metasurface grating design, and material absorption is always believed to be detrimental [8,12–22].

Distinct from those understandings [8,12–22], in this work, we propose and first experimentally demonstrate adjustment of absorption coefficients of the metasurface material as a new method to achieve uniform and efficient first-order diffraction among RGB primary colors. Physical mechanism is systematically investigated in theory and the feasibility of this method is well verified experimentally. A balanced first-order diffraction efficiency of approximately 20% is achieved for RGB colors across a FOV as large as ∼30° over a single-piece glass substrate. We believe the findings will advance our understanding of metasurface-based RGB displays, with implications for color balancing and efficient light coupling.

2. Results and discussion

2.1 Design and structure

As schematically shown in Fig. 1(a), a double-nanobeam grating made of absorptive a-Si is designed as a light in-coupling waveguide coupler to study the effect of the grating intrinsic absorption on the RGB diffraction efficiencies and we propose a new method of adjustment of a-Si absorption coefficients for equalizing the RGB efficiencies over a wide FOV. This double-nanobeam grating, compared with either a single-nanobeam or a three-nanobeam counterpart, can achieve more uniform efficiencies at RGB three colors over entire FOV, as analyzed in our previous work [22]. In Fig. 1(a), the grating has a period of p and thickness of t. In each period, there are two nanobeams with widths of w₁ and w₂ and separated by g.

 

Fig. 1. (a) Schematic diagram of the double-nanobeam metasurface grating for light in-coupling. (b) The +1st-order diffraction angle, α, as a function of the incident angle, θ, for all the RGB lights when the grating period p = 365 nm. When n_s = 1.96, the critical angle of total internal reflection is α_c = 30.68°, denoted as a black dotted line.

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RGB light shining from the top with an incident angle, θ, is diffracted by the grating into different orders in the glass substrate and the major energy will be coupled to the first diffraction order with an angle, α (Fig. 1(a)). The glass substrate serves as a waveguide for the +1st-order diffracted lights. It has higher refractive indices at RGB wavelengths than the surrounding air. To ensure lossless propagation of the +1st-order diffracted lights, α at each color is required to be greater than the critical angle, α_c, of the total internal reflection at the air/glass substrate according to the basic waveguide theory. Based on the grating equation and the requirement of total internal reflection of the RGB diffracted light, the maximum FOV and the grating period, p, can be determined [22]. The maximum FOV increases with the glass refractive index, n_s. When n_s = 1.96, a maximum FOV = 30.23° is obtained. In this case, 364.45 nm < p ≤ 365.42 nm is achieved. When we choose p = 365 nm, all the RGB diffraction angles are greater than α_c = 30.68° over the whole FOV (Fig. 1(b)). This guarantees that such a grating diffracts the incident RGB lights into the first diffraction orders and the diffracted lights propagate losslessly in the glass substrate. Further increase of n_s allows an even large FOV but will decrease the grating period [22]. As mentioned above, the p reduction will lead to the decrease of the first-order diffraction efficiency [8,11]. Therefore, in this work, we set p = 365 nm and n_s = 1.96 for the double-nanobeam metasurface grating on the glass substrate to present sufficiently high and balanced diffraction efficiencies over a sufficiently large FOV of 30° (i.e., [-15°, 15°]) unless otherwise specified. In this calculation and the following numerical simulation, the blue, green and red light wavelengths are set to λ₁ = 460 nm, λ₂ = 530 nm, and λ₃ = 620 nm, respectively, which are widely used in the field of LED illuminated micro-projector (e.g., RGB SMT modules from LE RTDUW S2W, LE ATB S2WWL or OSRAM).

2.2 Absorption coefficient of a-Si tailoring the grating diffraction efficiency balance among RGB colors

In order to demonstrate the impact of absorption coefficient on the diffraction efficiency balance among the RGB colors, two-dimensional full-vector numerical simulations were conducted based on a finite-difference time domain (FDTD) method using the commercial software Ansys Lumerical FDTD. Five groups of refractive indices, n + iκ, of a-Si were applied, which were obtained by measuring five a-Si film samples (Appendix A). As shown in Figs. 2(a1)–2(a5), the refractive indices vary among Cases 1-5. They have similar n values but gradually decreasing κ values from Case 1 to 5. For each case, geometric parameters of w₁, w₂, t, and g of the double-nanobeam gratings are optimized and the optimal values are summarized in Table 1. The optimization method was reported in detail previously [22].

 

Fig. 2. (a1-a5) Five groups of refractive indices, i.e., n + iκ (n: blue curves, κ: red curves) of a-Si chosen for Cases 1-5. (b1-b5) The first-order diffraction efficiencies at RGB wavelengths across FOV for the double-nanobeam metasurface gratings with the optimal geometric parameters for Cases 1-5.

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Table 1. The geometric parameters of the double-nanobeam metasurface gratings optimized with the a-Si refractive indices in Fig. 2(a1-a5) in Cases 1-5.

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For this in-coupling metasurface grating, only +1st-order diffraction efficiencies of the RGB colors across FOV are calculated and shown in Figs. 2(b1)–2(b5) for the optimal geometries in the five cases. The zero-order transmissions are not considered because in practical applications, the transmitted RGB lights are usually blocked by a black cover in AR glass assembly or reflected back into the in-coupling grating using a reflective coating on the other side of glass substrate to enhance overall efficiency. With the help of total internal reflection, only the +1st-order diffraction light beam is useful in AR glasses to accomplish 2-dimensional pupil expansions by one or two out-coupling gratings through zigzag beam propagation within glass substrate. Moreover, when the +1st-order diffraction is optimized and enhanced by our metasurface grating, the other diffraction orders, including the -1st-order diffraction, will be suppressed and can be neglected, like the behaviors of a conventional blazed grating [13]. Therefore, only the +1st-order diffraction is considered in this work, as in most present designs of AR glasses. Here the diffraction efficiency, η, is defined as the ratio of the +1st-order diffracted light power transmitted through the metasurface grating to the incident light power. In the numerical simulation, it is calculated with the equation below:

For Case 1, the +1st-order diffraction efficiencies are well balanced among the three colors over the entire FOV and change slightly with the incident angle, θ (Fig. 2(b1)). As the intrinsic absorption of a-Si decreases from Case 1 to Case 5, the RGB diffraction efficiencies increase while the RGB efficiency uniformity deteriorates, which is particularly noticeable at θ = 15° (Figs. 2(b1)–2(b5)). Specifically, the efficiency of the green light remains higher than that of the blue light and their difference at a given angle becomes larger with decreasing κ values. The efficiency of the red light almost follows that of the green light until θ increases to about 7°, where it starts to drop quickly and becomes far below those of the green and blue lights at θ = 15°, resulting in a significant divergence among RGB efficiencies.

Two parameters are defined to quantize the equalization of RGB efficiency: one is non-uniformity, η_NU, expressed in Equation (2), reflecting the maximum deviation of the RGB diffraction efficiencies, η’s, throughout the entire FOV; the other is average efficiency change, η_AC, defined by Equation (3), which characterizes the average change in the RGB diffraction efficiencies over the FOV.

Figure 3 shows the calculated η_NU and η_AC of the five cases. Both η_NU and η_AC increase monotonously with the decreasing imaginary parts of refractive indices. η_NU is well below 0.15 for Cases 1-3 with higher absorption coefficients (Fig. 3), showing a uniform RGB diffraction efficiencies (Figs. 3(b1)–3(b3)); while for Cases 4 and 5 with lower absorption coefficients, η_NU’s are well above 0.15 (Fig. 3), demonstrating poor uniformity of RGB diffraction efficiencies (Figs. 2(b4) and 2(b5)). A quite slow increasing rate is observed for η_AC from Case 1 to Case 4, whose η_AC values are well below 0.075 (Fig. 3), meaning a small variation among diffraction efficiencies of the three colors throughout the whole FOV (Figs. 2(b1)–2(b4)). Among the five cases, Case 5 with the minimum absorption coefficients features the highest value η_AC of above 0.1 (Fig. 3), which is consistent with the large variation among RGB diffraction efficiencies across FOV (Fig. 2(b5)). The above phenomena indicate that an absorptive material with certain κ values at RGB wavelengths is favorable for and can be employed to achieve uniform RGB diffraction efficiencies. Though the efficiencies are lowered by a little, overall system brightness may be compensated by an increased LED intensity.

 

Fig. 3. Calculated non-uniformity, η_NU, and average change, η_AC, of RGB +1st-order diffraction efficiencies across the FOV from -15° to 15° for Cases 1-5.

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2.3 Physical analysis

In order to understand the physics behind the above phenomena, we further investigated and analyzed the electric field, Poynting vector and phase distributions at the grating cross sections under RGB illumination at incident angles of θ = -15°, 0°, and 15°, respectively. Here Cases 1, 3, and 5 are representative and will be discussed in detail here. Understanding the optical behaviors in these cases helps us to understand the optical behaviors of all the five cases and the manipulation mechanism of the absorption coefficients on diffraction in a concise and clear way. On the other hand, Cases 2 and 4 will be described specifically below in Section 2.4 and will not be elaborated here. In a metasurface, subwavelength structures with high n values often serve as resonators for light field localization, associated with strong light scattering and large phase change, i.e., the wavefront change. If the subwavelength structures simultaneously have certain κ values, the localized resonances also suffer an absorption, which increases with the increasing κ values. Conventionally, the material absorption is often controlled to be as low as possible, but in this work we find that it is beneficial for RGB efficiency balance.

It can be seen that an asymmetrical grating structure under normal incidence results in an asymmetric distribution of light field in terms of both amplitude (Fig. 4) and phase (Fig. 5). In Fig. 4, the length of the white arrow represents the deflection amplitude of the Poynting vector, including light for ±1 and 0 and other higher orders of deflection. Portion of the arrows pointing to the lower right corner can be attributed to the light energy coupled to the +1st diffraction order of the grating. Using green light illumination as an example, a comparison between Fig. 4(b) with Fig. 4(h) shows that the beam works as a subwavelength resonator in the grating structure, and the resonance mainly occurs in the wider nano-strip, while the narrow nano-strip enhances deflection of Poynting vectors toward right. It is shown that under the same illumination, as the imaginary part of the refractive index decreases, i.e. absorption loss of the grating structure decreases, the resonance intensity increases as shown in Figs. 4(b), 4(e), and 4(h). Linking field strength to phase distributions, we observed a sharp phase change profile associated with a strong resonance (Figs. 4(h) and 5(h)), as well as discrete phase change domains pertained to three separate resonance regions (Fig. 5(g)) with relatively close resonance strength (Fig. 4(g)). For the case with the lowest absorption when the imaginary part of the refractive index is close to zero (Case 5), the resonance is the most significantly enhanced (Figs. 4(g), 4(h), and 4(i)), and the deflection of the wavefront is the largest (Figs. 5(g), 5(h), and 5(i)). For gratings with relatively high absorption coefficients (Cases 1 and 3), there is a competition between scattering and absorption caused by resonance, which jointly affect the first-order diffraction efficiency of incident light. For the grating with small absorption coefficients (Case 5), the absorption effect is weakened and scattering dominates. As shown in Figs. 2 and 3, Case 5 (Figs. 2(a5) and 2(b5)) has the highest energy coupled to the first-order diffraction, but exhibited a large deviation among efficiency of RGB primary color light (Fig. 3), while the smallest deviation among diffraction efficiency was observed in Case 1 (Figs. 2(a1) and 2(b1)) for RGB light (Fig. 3). Go search for the reason for the deviation, it is interesting to compare resonance domains in Figs. 4(g), 4(h), and 4(i), and also phase change domains in Figs. 5(g), 5(h), and 5(i). We see that short wavelength (blue light) tends to initiate sparse and small regional resonances, as shown in Fig. 4(g) and Fig. 5(g), whereas long wavelength (red light) causes resonance in a relatively spread region centered in wider nano-strip (Fig. 4(i)) and slow-varying phase distribution (Fig. 5(i)). From this point of view, it is mainly wavelength-dependent resonances that produce the difference in diffraction efficiency. It is shown in either Case 1 or 3 that this discrepancy in the efficiency may be suppressed effectively by introducing a suitable amount of medium absorption.

 

Fig. 4. Electric field intensity (|E|²) distributions over the cross-sections of the gratings in (a-c) Case 1, (d-f) Case 3, and (g-i) Case 5 under normal incidence (θ = 0°) of lights at (a,d,g) λ₁ = 460 nm, (b,e,h) λ₂ = 530 nm, and (c,f,i) λ₃ = 620 nm. The time-averaged Poynting vectors are indicated as white arrows over the electric field intensity distributions. In each inset, the unit cell structure of the grating is outlined by grey dashed lines.

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Fig. 5. Phase (φ_E) distributions over the cross-sections of the gratings in (a-c) Case 1, (d-f) Case 3, and (g-i) Case 5 under normal incidence (θ = 0°) of lights at (a,d,g) λ₁ = 460 nm, (b,e,h) λ₂ = 530 nm, and (c,f,i) λ₃ = 620 nm. In each inset, the unit cell structure of the grating is outlined by grey dashed lines.

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Under the oblique incidence at θ = -15°, with the decreasing imaginary part of the refractive index through Case 1 to 5, the resonance is enhanced and light scattering into the first-order diffraction increases (Fig. 6), similar to the case of normal incidence (Fig. 4). When θ changes from 0° to -15°, both the light fields and phase contours rotate clockwise (Fig. 6 v.s. Fig. 4, Fig. 7 v.s. Fig. 5), following the change of θ; for gratings with higher absorption (Cases 1 and 3), the resonance locations, intensity (Fig. 6) and phase distribution (Fig. 7) change little for RGB colors, and their diffraction efficiencies remain to be nearly the same as those of the normal-incidence cases under the combined effects of local resonance-induced absorption and scattering. For gratings with minimal absorption (Case 5), at blue light, the resonance occurs mainly at two corners in nano-strips instead of three regions (Fig. 6(g) v.s. Fig. 4(g)), and the phase change is more concentrated near the two nanobeams (Fig. 7(g) v.s. Fig. 5(g)), due to change in effective cross-section of subwavelength structure toward incoming light waves at this incident angle, leading to more energy coupled to the +1st-order diffraction despite the relatively higher absorption coefficient of a-Si than those at the green and red colors. At green and blue lights, the larger wavelengths make phase profiles almost unchanged except clockwise rotation relative to those of the normal incident cases (Figs. 7(h) and 7(i) and Figs. 5(h) and 5(i)). With reduced absorption, their resonance intensities (Figs. 6(h) and 6(i)) become stronger than that of the blue light. However, more energy is diffracted to higher-order diffractions (Figs. 6(h) and 6(i)), resulting in a degradation in efficiency of the +1st diffraction order. Therefore, a convergence of diffraction efficiencies of the three primary colors is observed (Fig. 2(b5)).

 

Fig. 6. Electric field intensity (|E|²) distributions over the cross-sections of the gratings in (a-c) Case 1, (d-f) Case 3, and (g-i) Case 5 under oblique incidence (θ = -15°) of lights at (a,d,g) λ₁ = 460 nm, (b,e,h) λ₂ = 530 nm, and (c,f,i) λ₃ = 620 nm. The time-averaged Poynting vectors are indicated as white arrows over the electric field intensity distributions. In each inset, the unit cell structure of the grating is outlined by grey dashed lines.

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Fig. 7. Phase (φ_E) distributions over the cross-sections of the gratings in (a-c) Case 1, (d-f) Case 3, and (g-i) Case 5 under oblique incidence (θ = -15°) of lights at (a,d,g) λ₁ = 460 nm, (b,e,h) λ₂ = 530 nm, and (c,f,i) λ₃ = 620 nm. In each inset, the unit cell structure of the grating is outlined by grey dashed lines.

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At the angle of incidence of θ = 15°, both the light field and phase contours rotate anticlockwise compared to the case under normal incidence (Fig. 8 v.s. Fig. 4; Fig. 9 v.s. Fig. 5). For Cases 1 and 3, due to the well-balanced scattering and absorption induced by the resonance, the first-order diffraction efficiencies of the blue and green lights do not degrade much compared to those of the normal incidence cases (Figs. 2(b1) and 2(b3)). For Case 5, at blue, the resonance introduces strong absorption overwhelming the induced scattering due to the large absorption coefficient (Poynting vector in Fig. 8(g) and phase or wavefront in Fig. 9(g)); oppositely, at green, the absorption coefficient is small and the resonance introduces strong scattering overwhelming the induced absorption (Poynting vector in Fig. 8(h) and phase or wavefront in Fig. 9(h)). Therefore, the diffraction efficiency at blue decreases but the efficiency at green increases as θ changes from 0° to 15° (Fig. 2(b5)). For Cases 1, 3 and 5, red light causes resonance over a large volume (especially for Case 5 in Fig. 8(i)), wherein absorption loss may probably exceed increase in diffraction efficiency due to the enhanced resonance despite the rather small absorption coefficient. Simultaneously, the phase changes greatly and both the wavefront and the Poynting vector point in a horizontal and even upward direction (Figs. 9(c), 9(f), and 9(i); Figs. 8(c), 8(f), and 8(i)), further reducing the energy diffracted into the first order. Therefore, the diffraction efficiency of the red light is much lower than those of the blue and green lights for the three cases (Figs. 2(b1), 2(b3) and 2(b5)). For Case 5, when θ increases from 0 to 15°, the efficiency first increases and then decreases, peaking at θ ≈ 7°(Fig. 2(b5)), indicating that resonance first enhances light scattering and thus the diffraction efficiency in the domain of 0° < θ < 7° and then in the domain of θ > 7° the relatively large volume resonance leads to not only great absorption but also incoming red light scattering toward upper right, as displayed by Poynting vectors in upper right corner of Fig. 8(i), and thus a quick decrease in efficiency. This angular dependence of resonance and associated backscattering also implies a structural drawback of a simple double nano-strip, which may be further modified by adding three-dimensional features to improve flattened wideband RGB efficiency.

 

Fig. 8. Electric field intensity (|E|²) distributions over the cross-sections of the gratings in (a-c) Case 1, (d-f) Case 3, and (g-i) Case 5 under oblique incidence (θ = 15°) of lights at (a,d,g) λ₁ = 460 nm, (b,e,h) λ₂ = 530 nm, and (c,f,i) λ₃ = 620 nm. The time-averaged Poynting vectors are indicated as white arrows over the electric field intensity distributions. In each panel, the unit cell structure of the grating is outlined by grey dashed lines.

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Fig. 9. Phase (φ_E) distributions over the cross-sections of the gratings in (a-c) Case 1, (d-f) Case 3, and (g-i) Case 5 under oblique incidence (θ = 15°) of lights at (a,d,g) λ₁ = 460 nm, (b,e,h) λ₂ = 530 nm, and (c,f,i) λ₃ = 620 nm. In each panel, the unit cell structure of the grating is outlined by grey dashed lines.

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For the equalization of RGB coupling efficiency in entire FOV, the absorption coefficient of the metasurface material plays an important role in the homogenization of RGB diffraction efficiency. The above analysis shows that a small imaginary part of the grating material index improves the first-order diffraction efficiency. Low loss resonance in many cases benefits efficient coupling of light into the first-order diffraction, but resonance-induced scattering also causes backscattering and leads to increased disparity among RGB tricolor efficiencies. The two effects produced by resonance, which are enhanced scattering and absorption, cause the RGB efficiency to be uneven in the FOV window and separate from one another. When the absorption coefficient of the metasurface medium is increased, scattering and absorption compete and balance each other, so an equalization of RGB tricolor diffraction efficiency can be achieved. Theoretically, a balanced and efficient first-order diffraction efficiency of all RGB light can be achieved in the FOV by selecting the appropriate complex refractive index.

2.4 First experimental demonstration

To verify the theoretical prediction of the aforementioned metasurface grating waveguide coupler, two waveguide couplers were prepared on two pieces of glass (SCHOTT RealView 1.8) and characterized with available lasers at 460, 524, and 655 nm. There was difference in light wavelength between experiment and numerical simulation, which however did not affect the demonstration of the absorption coefficient effect on the balance of the RGB diffraction efficiencies (to be demonstrated below). The a-Si film thicknesses of Samples 1 and 2 were measured to be 65 and 90 nm, respectively. Their complex refractive indices, i.e., n + iκ, were derived analytically from the measured reflectivities and transmissivities and plotted in Figs. 10(a) and 10(b), together with the refractive indices of the 5 cases considered in the above theoretical analysis. It is shown that the derived n and κ values are well within the range spanned by those of 5 cases. Specifically, the imaginary parts of the indices of the a-Si films of Samples 1 and 2 match well with those of Cases 2 and 4, respectively (Fig. 10(b)). Considering the dominant role of the absorptive coefficient on the diffractive efficiency as discussed above, the two coupler samples were fabricated according to the optimized geometric parameters of Cases 2 and 4, respectively (Table 1). The details about film deposition, coupler fabrication and characterization can be found in the Appendix A.

 

Fig. 10. (a) Real parts, n, and (b) imaginary parts, κ, of refractive indices of a-Si films of Samples 1 and 2 as well as of Cases 1-5 discussed in the theoretical analysis. (c) SEM image of Sample 2.

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Figure 10(c) shows one representative scanning electron microscopy (SEM) image of Sample 2, where the narrow and wide nanobeams can be clearly observed in spite of a certain degree of line edge roughness (LER). The geometric parameters of the nanostructures by SEM inspection are summarized in Table 2. Compared with the numerically optimized geometric parameters of Cases 2 and 4 listed in Table 1, thicknesses, widths and gaps of Samples 1 and 2 were deviated from the desired values due to challenges such as beam blur and sidewall etching in fabrication of nanostructures with a critical dimension of tens of nanometers. The narrow nanobeam was wider, while the wide nanobeam was narrower than design values. And spacing between them turned to be wider, especially for Sample 2. Both the grating period and thickness were greater than the optimal values. These deviations between samples and optimal design are mainly caused by the limit of lithographic fabrication processes available to us. The proximity effect of the electron-beam lithography (EBL) affects the size accuracy. The LER of the sub-100 nm wide nanobeam is practically difficult to avoid by EBL and inductively coupled plasma (ICP) etching. Nevertheless, desired measurement results are still obtained below.

Table 2. The geometric parameters of the fabricated double-nanobeam metasurface gratings obtained from SEM and AFM inspections. The parameters are averaged.

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With the setups schematically shown in Fig. 11(a), diffraction efficiencies of Samples 1 and 2 were measured (Appendix A) and plotted in Figs. 11(c) and 11(d), respectively. For comparison, we simulated the theoretical diffraction efficiencies by using the actual refractive indices (Figs. 10(a) and 10(b)) and the actual geometric parameters (Table 2) in the numerical model and plotted them in Figs. 11(e) and 11(f). Note that the structural parameters in Table 2 cannot fully reflect the real structures because of the non-uniformity of the fabrication and that the rough sidewall of the nanobeams are neglected in the numerical model. Therefore, the theoretical results do not fully match the experimental results. Nevertheless, both the experimental and simulation results show that due to the larger absorptive coefficients, Sample 1 demonstrates more uniform diffraction efficiencies among RGB tricolors than Sample 2, well verifying the above theoretical analysis. Besides, RGB diffraction efficiencies of approximately 20% were achieved over FOV as large as ∼30°, showing very good efficiency uniformity. As far as we know, it is the first time for us to experimentally demonstrate the potential of using only a single-piece AR glass plate to realize balanced RGB diffraction efficiencies. Note that there is optical dispersion of the glass substrate with an average refractive index of about 1.8 in the visible regime (Fig. 12(a), Appendix B), different from the value set in the above theoretical analysis. However, it is found that the glass index has little influences on both the RGB +1st-order diffraction efficiencies of the grating coupler over the whole incident angle range and efficiency balance among RGB three colors (Appendix B).

 

Fig. 11. (a) Schematic diagram of the optical characterization setups. (b) Calculated non-uniformity, η_NU, and average change, η_AC, of both measured (solid stars) and simulated (hollow circles) RGB +1st-order diffraction efficiencies across the FOV from -15° to 15° for Samples 1 and 2. The results for Cases 1-5 with optimal geometrical parameters are also displayed for comparison. The results of Samples 1 and 2 are plotted over those of Cases 2 and 4, respectively. (c,d) Measured and (e,f) simulated +1st-order diffraction efficiencies versus incident angle, θ, for Samples (c,e) 1 and (d,f) 2. The geometric parameters listed in Table 2 and the derived refractive indices in Figs. 10(a) and 10(b) were utilized in the simulation.

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Fig. 12. (a) Refractive index of the glass substrate, n_s, used in our experimental demonstration. Calculated RGB +1st-order diffraction efficiencies of (b,c) Sample 1 and (d,e) Sample 2 (b,d) without and (c,e) with glass dispersion.

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We calculated the non-uniformity, η_NU, and average change, η_AC, of both measured and simulated RGB +1st-order diffraction efficiencies over the FOV from -15° to 15° for Samples 1 and 2 with Equations (2) and (3) and plotted them in Fig. 11(b). Here those results for Cases 1-5 with optimal geometric parameters are also shown for comparison, and the calculated results of Samples 1 and 2 are plotted over those of Cases 2 and 4, respectively, for their well-matched absorption coefficients as shown in Fig. 10(b). It is seen that both η_NU and η_AC obtained from the experimentally measured efficiencies follow the trends of those calculated from simulated efficiencies. The experimental results are relatively larger due to the non-perfect fabrication processes. Sample 1 has a smaller η_NU than Sample 2 (experiment: 0.17 v.s. 0.21; simulation: 0.07 v.s. 0.15), consistent with the optimal cases, i.e., Cases 2 and 4. A larger η_AC is observed for Sample 1 rather than Sample 2 (experiment: 0.1 v.s. 0.07; simulation: 0.08 v.s. 0.07). This is only slightly different from the optimal cases where η_AC for Case 4 is slightly higher than that for Case 2.

To investigate whether the incident angle range can be extended, the +1st-order diffraction efficiency of Sample 1 is further simulated numerically when the RGB incident angle, θ, ranges from -50° to 50°. η_NU and η_AC are calculated over different extended θ ranges, i.e., [-20°, 20°], [-30°, 30°], [-20°, 15°] and [-30°, 15°], which are compared with the values calculated over [-15°, 15°]. It is found that the best working angle range is still from -15° to 15° with the smallest η_NU and η_AC values. Details can be found in Appendix C.

3. Conclusion

In this study, we proposed adjustment of absorption coefficient as a new method for equalization of RGB diffraction efficiencies over a wide FOV. It is found that both η_NU and η_AC increase monotonously with the decreasing imaginary parts of refractive indices. For Cases 1-3 with higher absorption coefficients, η_NU and η_AC are well below 0.15 and 0.075, respectively, representing a good uniformity and angle-insensitive RGB diffraction efficiencies over a wide FOV of 30°. The systematic analysis revealed crucial role of both resonance-induced light scattering and absorption in efficient and balanced RGB diffraction coupling. Further we first experimentally demonstrated the feasibility of our method and achieved RGB efficiencies of approximately 20% across a FOV as large as ∼30° over a single-piece glass plate with η_NU ∼ 0.17 and η_AC ∼ 0.1. These findings advance our understanding of optimizing RGB waveguide couplers, with implications for color balancing and efficient light coupling. Future research could explore additional design variations and optimization strategies to further enhance the performance of RGB waveguide couplers.

Appendix A: methods

Film deposition and characterization

To obtain the complex refractive indices, i.e., n + iκ, in Cases 1-5, five a-Si films were deposited with an electron-beam evaporator followed by a thermal annealing treatment. The crystallinity of the a-Si film was sensitive to the annealing temperature and time. Any small changes in the annealing condition would cause quite different crystallinities and thus different refractive indices of the a-Si film. The n and κ values of the films were measured with an ellipsometer.

For the two metasurface grating waveguide couplers, i.e., Samples 1 and 2, the a-Si films were deposited and treated with the above methods on the front surfaces of the 0.7-mm thick glass substrates with refractive indices of about 1.8 in the visible regime (SCHOTT RealView 1.8). There were anti-reflective layers (ARLs) on the back surfaces. To obtain the complex refractive indices of the two a-Si films, two a-Si/glass/ARLs multi-layer structures were characterized with lasers at wavelengths of 460, 524, and 655 nm and an incident angle of 8°. Their reflectivities and transmissivities were measured and introduced into a transfer matrix method (TMM) model built with the commercial software Ansys Lumerical FDTD. Since the details of the ARLs were unclear, to simplify the TMM analysis, a single MgF₂ ARL was assumed with refractive index of 1.38 and its thickness was determined to enable minimal reflectivities at 460, 524, and 655 nm. Then the a-Si/glass/MgF₂ three-layer structure was built and analyzed with the TMM method. For each light wavelength, n and κ were scanned simultaneously to calculate reflectivities and transmissivities. When the calculated values were the closest to the measured values, the n and κ values corresponding to the calculated reflectivities and transmissivities were recorded as real values.

Coupler fabrication and characterization

Samples 1 and 2 were fabricated according to the optimal geometric parameters of Cases 2 and 4, respectively. Briefly a thin conductive layer and a photoresist layer of ZEP 520 were spin-coated in sequence on top of the a-Si film for EBL (Raith EBPG 5200). The conductive layer was applied to improve the conductivity of the a-Si film and the resolution of EBL. After development, the grating pattern was formed in the resist layer. With the protection of the resist, the a-Si film was etched using ICP etching (Oxford Instruments). Finally, the residual resist was removed by acetone cleaning, and the double-nanobeam grating was formed.

The surface morphologies of the two samples were inspected with a SEM and the geometric parameters were extracted. The a-Si thicknesses of Samples 1 and 2 were measured with an atomic force microscope. Optical characterization was performed with the setup schematically shown in Fig. 11(a). A co-line RGB laser module was utilized to generate collimated RGB beams with polarization aligned with sample grating and wavelengths at 460, 524, and 655 nm. Metasurface structure on the glass substrate was placed on center axis of a semi-cylindrical lens, with index matching fluid between sample and lens. The sample and lens assembly were on a rotation stage allowing laser beam to incident onto sample with variable angles. The output light power (I_o1) of the +1st-order diffraction from the semi-cylindrical lens was detected and normalized to the output power of light incident on the bare glass without the grating (I_o2). The normalized value, i.e., I_o1/I_o2, was recorded as the +1st-order diffraction efficiency.

Appendix B: effect of the refractive index of the glass substrate

For the glass substrate used in our experimental demonstration, there is optical dispersion. As shown in Fig. 12(a), its refractive index, n_s, decreases as the light wavelength, λ, increases.

To investigate the effect of the glass dispersion, we conducted numerical simulations of Samples 1 and 2 with and without glass dispersion. For the samples without glass dispersion, the refractive index of glass was set to 1.8, i.e., n_s = 1.8. For the samples with glass dispersion, the refractive indices in Fig. 12(a) were introduced in the numerical model. The a-Si refractive indices and the geometric parameters of Samples 1 and 2 were set to the measured values, as shown in Fig. 10 and Table 2, respectively. The RGB +1st-order diffraction efficiencies of the two samples with and without glass dispersion were calculated under an incident angle, θ, from -20° to 20° and plotted in Figs. 12(b)-12(e). It is clearly seen that for both Samples 1 and 2, the calculated RGB diffraction efficiencies of the cases with and without the glass dispersion are nearly indistinguishable. Better efficiency equalization is observed for Sample 1 either with or without glass dispersion rather than for Sample 2.

Appendix C: maximum working angle of the incident light

As mentioned in the main text, for Sample 1, good efficiency equalization is achieved when the incident angle, θ, ranges from -15° to 15° (Figs. 11(c) and 11(e)). Over this angle range, the non-uniformity, η_NU, is 0.17 and 0.07 in experiment and simulation, respectively; the average efficiency change, η_AC, are 0.1 and 0.08 in experiment and simulation, respectively.

To investigate the maximum working angle of the incident light, the +1st-order diffraction efficiency of Sample 1 is simulated numerically when the RGB incident angle, θ, ranges from -50° to 50°, as shown in Fig. 13(a). The +1st-order diffraction disappears when θ ≥ 48°, 37°, and 18° respectively for the blue, green and red light. Near these cutoff angles, the diffraction efficiency drops dramatically to zero. Therefore, if the θ range is expanded beyond 30°, the RGB balance will be broken. As shown in Fig. 13(b), when the θ range is extended from [-15°,15°] to [-20°,20°], η_NU increases from 0.07 to 0.14, and η_AC increases from 0.08 to 0.12. When the θ range is further extended to [-30°,30°], η_NU = 0.14 keeps but η_AC continues to rise to 0.14. On the other hand, if we keep the maximum θ value as 15° and varies the minimum θ value to e.g., -20° and -30°, η_NU and η_AC increase moderately to 0.09 and 0.09, respectively. As θ decreases from -15°, the diffraction angles of the RGB lights also decrease. For each color, once the diffraction angle is smaller than the critical angle at the air/glass interface, propagation loss occurs along the waveguide. Though the diffraction efficiency balance seems not be too bad, the light intensity arriving at the eye might not equalize if the in-coupling grating is used as the out-coupling grating. In this case, good visual experience cannot be achieved by the AR wearers. Therefore, the best working angle range is still from -15° to 15°.

 

Fig. 13. (a) Calculated RGB +1st-order diffraction efficiencies of Sample 1 when the incident angle, θ, ranges from -50° to 50°. (b) Calculated non-uniformity, η_NU, and average change, η_AC, of RGB +1st-order diffraction efficiencies when θ ∈ [-15°,15°], [-20°,20°], [-30°,30°], [-20°,15°] and [-30°,15°].

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Funding

Grant from ZJU-Sunny Photonics Innovation Center (2020-06); National Key Research and Development Program of China (2017YFA0205700); National Natural Science Foundation of China (61307078, 61775195).

Disclosures

The authors declare no conflicts of interest.

Data availability

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

References

1. Z. E. Bomzon, G. Biener, V. Kleiner, et al., “Space-Variant Pancharatnam-Berry Phase Optical Elements with Computer-Generated Subwavelength Gratings,” Opt. Lett. 27(13), 1141–1143 (2002). [CrossRef]  

2. D. Lin, P. Fan, E. Hasman, et al., “Dielectric Gradient Metasurface Optical Elements,” Science 345(6194), 298–302 (2014). [CrossRef]  

3. N. F. Yu, P. Genevet, M. A. Kats, et al., “Light Propagation with Phase Discontinuities: Generalized Laws of Reflection And Refraction,” Science 334(6054), 333–337 (2011). [CrossRef]  

4. X. G. Ni, N. K. Emani, A. V. Kildishev, et al., “Broadband Light Bending with Plasmonic Nanoantennas,” Science 335(6067), 427 (2012). [CrossRef]  

5. F. Monticone, N. M. Estakhri, and A. Alù, “Full Control of Nanoscale Optical Transmission with a Composite Metascreen,” Phys. Rev. Lett. 110(20), 203903 (2013). [CrossRef]  

6. C. Pfeiffer and A. Grbic, “Metamaterial Huygens’ Surfaces: Tailoring Wave Fronts with Reflectionless Sheets,” Phys. Rev. Lett. 110(19), 197401 (2013). [CrossRef]  

7. M. Decker, I. Staude, M. Falkner, et al., “High-Efficiency Dielectric Huygens’ Surfaces,” Adv. Opt. Mater. 3(6), 813–820 (2015). [CrossRef]  

8. D. Lin, M. Melli, P. St. Hilaire, et al., “Metasurfaces with Asymmetric Gratings for Redirecting Light and Methods for Fabricating,” U.S. patent, US10,527,851B2 (7 January 2020).

9. Z. Li, P. Lin, Y. Huang, et al., “Meta-Optics Achieves RGB-Achromatic Focusing for Virtual Reality,” Sci. Adv. 7(5), eabe4458 (2021). [CrossRef]  

10. M. G. Moharam and T. K. Gaylord, “Diffraction Analysis of Dielectric Surface-Relief Gratings,” J. Opt. Soc. Am. 72(10), 1385–1392 (1982). [CrossRef]  

11. M. W. Farn, “Binary Gratings with Increased Efficiency,” Appl. Opt. 31(22), 4453–4458 (1992). [CrossRef]  

12. B. Kress, “Design, Modelling and Fabrication Techniques for Micro-Optics,” SPIE SC 1125, SPIE PW19, SF.

13. P. Lalanne, S. Astilean, P. Chavel, et al., “Design and Fabrication of Blazed Binary Diffractive Elements with Sampling Periods Smaller Than the Structural Cutoff,” J. Opt. Soc. Am. A 16(5), 1143–1156 (1999). [CrossRef]  

14. T. Levola, “Novel Diffractive Optical Components for Near to Eye Displays,” Dig. Tech. Pap. - Soc. Inf. Disp. Int. Symp. 37(1), 64–67 (2006). [CrossRef]  

15. T. Levola and P. Laakkonen, “Replicated Slanted Gratings with a High Refractive Index Material for In and Outcoupling of Light,” Opt. Express 15(5), 2067–2074 (2007). [CrossRef]  

16. T. Levola, “Steroscopic Near to Eye Display Using a Single Microdisplay,” Dig. Tech. Pap. - Soc. Inf. Disp. Int. Symp. 38(1), 1158–1159 (2007). [CrossRef]  

17. T. Levola and V. Aaltonen, “Near-to-Eye Display with Diffractive Exit Pupil Expander Having Chevron Design,” J. Soc. Inf. Disp. 16(8), 857–862 (2008). [CrossRef]  

18. Z. Zhou, J. Li, R. Su, et al., “Efficient Silicon Metasurfaces for Visible Light,” ACS Photonics 4(3), 544–551 (2017). [CrossRef]  

19. Y. Amitai, “Extremely Compact High-Performance HMDs Based on Substrate-Guided Optical Element,” Dig. Tech. Pap. - Soc. Inf. Disp. Int. Symp. 35(1), 310–313 (2004). [CrossRef]  

20. J. Han, J. Liu, X. Yao, et al., “Portable Waveguide Display System with a Large Field of View by Integrating Freeform Elements And Volume Holograms,” Opt. Express 23(3), 3534–3549 (2015). [CrossRef]  

21. Z. Liu, Y. Pang, C. Pan, et al., “Design of a Uniform-Illumination Binocular Waveguide Display with Diffraction Gratings and Freeform Optics,” Opt. Express 25(24), 30720–30731 (2017). [CrossRef]  

22. D. Gu, C. Liang, L. Sun, et al., “Optical Metasurfaces for Beam Steering with Uniform Efficiencies at RGB Wavelengths,” Opt. Express 29(18), 29149–29164 (2021). [CrossRef]