1. Introduction

Augmented reality (AR) technology provides an immersive experience that seamlessly integrates information from the virtual world with that of the real world [1]. With the increased computing power of on-the-go electronics, mobile devices with AR capabilities are becoming the next generation computing platforms [2,3]. As one of the key components of the see-through AR head-mounted display (HMD) [4], the optical waveguide [5] transfers the images produced by the micro-display into the visual frame of human eyes. The optical waveguides can be categorized into array waveguides [1,6] and diffractive waveguides [7–9] based on the way how the light is coupled in and out.

The array waveguide is built on the principle of cascaded mirrors with partially reflective coatings to achieve image conduction and eye-box expansion [10,11]. It has the advantages of high brightness, simple design and easy manufacture [12]. By 2021, Lumus Ltd. further enhanced the competitiveness of the array waveguide by a 2-D array design, which increases the diagonal field of view (DFOV) from 20° to 50° [13]. However, the array waveguides that are formed by high precision coating, cutting, bonding and polishing processes have a low tolerance for parallelism errors, which tends to cause stray light and ghost images [12,14]. Diffractive waveguide enables 2D expansion and image display by means of the small image generator, in-coupling grating and exit pupil expander(s) [15]. Compared to the array configuration, diffractive waveguides are prone to issues with low/uneven brightness and clarity of the virtual images, but they require only a cost-effective nanoimprinting process to form grating structures on the surface of waveguides [16,17]. The technology offers a good balance between form factor, FOV and size of eye-box and thus is receiving considerable attention from both scientific and industry communities [9].

The diffractive waveguide system consists of in-coupling diffractive optical element (DOE), total internal reflection (TIR) optical waveguide, and out-coupling DOE [4]. The function of in-coupling DOE is to couple the light generated by the micro-display into the optical waveguide. Depending on the diffraction characteristics of in-coupling DOE, diffracted light generated by incidence from different angles and wavelengths are propagated in the waveguide by TIR along their respective directions. The direction and efficiency of diffracted light can be optimized by adjusting the structural parameters such as period, duty cycle (DC) and height (H) [18]. The out-coupling DOE breaks the TIR conditions, allowing the light beam to leak out of the optical waveguide and into the eyes. Based on the working principle of the diffraction waveguide described above, the in-coupling DOE, which guides incident light into the first-order diffraction directions across angular ranges of FOV, plays a key role in determining the utilization of the light source and image quality [19,20].

One-dimensional (1-D) slanted, triangle and binary gratings have been proposed as in-coupling DOEs. Slanted [21–24] and triangle [23,25] gratings exhibit a tilted and asymmetric structure that concentrate most of incident light into first-order diffraction direction, increasing in-coupling diffraction efficiency (η). More than 80% of the first-order η can be obtained on these asymmetric gratings [23,26], while the challenges on master mold fabrication and demolding process in nanoimprinting limit their product yields and industrialization [27]. Compared to slanted and triangle gratings, the symmetric binary gratings feature lower efficiency in first-order diffraction, but are most commonly used in practice because of their simple processing [4]. Taking advantage of the symmetric diffraction, binary gratings are also proposed to be used in binocular diffraction waveguide system that enables the left and right eyes to receive the image from one micro-display [20].

In a practical diffractive AR system, the in-coupling DOE, which is usually a few millimeters away from the exit pupil position of the micro-display, should be somewhat larger than the exit pupil of micro-display to ensure that all incident light at a given FOV can be coupled into the waveguide. The exit pupil diameter of DLP (digital light processor) and LCoS (liquid crystal on silicon) micro-displays [28] is usually 4 mm, so the size of the in-coupling DOE should be larger than 4 mm. Moreover, assuming a diffracted light with an angle of 50° and a waveguide thickness of 0.6 mm, the distance along the horizontal direction between two adjacent TIRs is around 0.715 mm. At this point, the diffracted light has the probability of making multiple interactions with the in-coupling DOE, resulting in energy loss by the back-coupled light [4], which is termed as back-coupling loss (BCL) in this work. At present, the scenario to improve the first-order diffraction efficiency without considering the BCL is apparently insufficient for realistic AR applications. And what we really concern is the diffracted light that can be propagated and reach the out-coupling DOE in the waveguide.

In this work, the power of light arriving at the out-coupling DOE (P_out) divided by the power of incident light (P_in) is denoted as optical efficiency in waveguide (OEW). The BCL is firstly studied on binary grating-based in-coupling DOEs, in which the OEW is used as the criterion for evaluating the in-coupling performance. As expected, more BCLs occur at negative angles of incident light due to the smaller diffraction angle and multiple light/grating interactions. A thin-film interlayer with lower or higher refractive indices is introduced between the in-coupling DOE and waveguide, and is demonstrated as an effective approach to improve the first-order η. Subsequently, the average OEW and its uniformity are uplifted from 8.02% and 24.83% to 8.34% and 35.02%, by adding a MgF₂ thin-film interlayer at a specific region. The proposed OEW criterion and interlayer scheme provide a new perspective for the design of efficient diffractive optical waveguide systems.

2. Methods

2.1. Parameters of the diffractive optical waveguides

Referring to the information on products of AR displays with diffractive waveguides, the refractive index and thickness of the waveguide are set to be 1.75 and 0.6 mm, respectively. The refractive index enables a DFOV of 40° (20° in x-axis and 35° in y-axis), while the thickness is a trade-off between comfort, processability and subsequent out-coupling efficiency. A monochromic light (532 nm) is employed as the light source that irradiates onto the transmission 1-D grating with an area of 5 × 7 mm² (5 mm in x-axis and 7 mm in y-axis). The groove of grating is vertical to the x-axis, which means the diffracted light will propagate along the x-axis direction. The grating period is determined by using the analysis of the wave vector (k) [19]. It is stated that the k propagating in a waveguide must satisfy the following equations:

2.2. Simulation methods for diffractive optical waveguides

VirtualLab Fusion software [29] based on Fourier Modal Method (FMM) and ray tracing method is employed to design in-coupling DOEs, where the FMM is used to calculate the diffraction angles along with their efficiencies, while the ray tracing method is used to study the light propagation properties. As shown in Supplement 1, Fig. S1, two simulation models are applied to calculate the BCL and OEW respectively. The in-coupling DOEs in both models have the same size (5 × 7 mm²) with the light source 4.5 mm away from the in-coupling DOE along its positive z-axis direction.

The field around the waveguide is set to Field is Absorbed to imitate the light-shielding film in real AR glasses, which avoids the light reflected back from the boundary coupling with the grating again. In the study of BCL effect (Supplement 1, Fig. S1a), the size of the light source is set to be 100×100 µm² to conveniently observe the ray paths. A Power detector is placed above the light source with an area size of 8×10 mm² to ensure that all BCLs are collected. In the evaluation of OEW, the size of the light source is set to be 3.4×4 mm². The out-coupling DOE with the same period as the in-coupling DOE is located at 10 mm (central position) along the x-axis (See Supplement 1, Fig. S1b), which serves to couple the diffracted light out of the waveguide in the same direction as the incident light. The efficiencies of out-coupling DOE are set to be 100% regardless the light direction. The OEWs are monitored by a Power detector located at 4 mm in the positive z-axis direction.

2.3 Rigorous coupled wave analysis method for electric field analysis

The electric field distributions of gratings are studied using rigorous coupled wave analysis (RCWA) method [30]. For the purpose of observing the interaction of multiple diffraction fields, the region consisting of five grating periods (along x-axis) is used as the simulation unit. The layers of the simulation model along the positive z-axis are waveguide, MgF₂ interlayer (if applicable), residual resin and binary grating in sequence. The grating period, DC and H are set as 390 nm, 40% and 100 nm. The MgF₂ thin-film (200 nm) is used as an interlayer upon which is the residual resin with a thickness 700 nm, while the thickness of the residual resin for the model without interlayer is set to 900 nm. A plane wave at 532 nm with an incident angle of (−5°, 0°) located above the grating is employed as the light source that irradiates onto the grating.

3. Results and discussions

Figure 1(a) schematically shows a diffractive AR system with in-coupling and out-coupling DOEs. The use of glass with a refractive index (n) of 1.75 allows for a DFOV of 40° in an AR system [19], which is comparable to the parameters of the current commercial products [31]. This work focuses on the transmission binary gratings as in-coupling DOEs in AR systems, in which the refractive index and period are set as 1.75 and 390 nm, respectively. To facilitate the description of angles, FOV values in the x- and y-axes are used to define the angle of a light beam, denoted as (FOV_X, FOV_Y), which range from −10° to 10° and −17.5° to 17.5°, and determine the DFOV of 40° (Supplement 1, Fig. S2).

 

Fig. 1. (a) A schematic diagram of an AR system with in-coupling and out-coupling DOEs. (b) FoM as a function of grating geometries where the highest value is marked by the black pentagram. (c) The distribution of η_T+1 with angle at H of 150 nm and DC of 65%.

Download Full Size | PDF

Herein, the average +1^st order efficiency (η_{T+1_ave}) of transmission binary grating is calculated as the sum of +1^st order efficiency (η_T+1) divided by the number of incident plane waves, while the uniformity is expressed as the minimum η_T+1 divided by η_{T+1_ave} [32]. The grating’s figure of merit (FoM_grating), determined as the product of the normalized η_{T+1_ave} ($_{\textrm{T} + {1_{\textrm{ave}}}}^{\prime}{ = _{\textrm{T} + 1\_\textrm{ave}}}{/_{\textrm{T} + 1\_\textrm{ave}\_\textrm{max}}}$) and uniformity, is used to evaluate the performance of in-coupling DOE, where the η_{T+1_ave_max} is the maximum value of η_{T+1_ave} of the gratings with different DC and H. In the subsequent optimization, the H (100∼500 nm) and DC (30∼70%) of the binary gratings are varied, while the period (390 nm) and refractive index (1.75) are fixed. The number of incident light is set to be 5×8 rays, i.e., FOV_X and FOV_Y are equally divided into 5 and 8 angles respectively. For each H and DC, the forty rays are simulated resulting in the η_{T+1_ave} and its uniformity as a function of grating geometries (See Supplement 1, Fig. S3). The η_{T+1_ave} is generally higher in the H and DC ranges of 150∼250 nm and 30∼55% respectively, while the higher uniformity is mostly located at the corresponding low efficiency regions. Considering the FoM_grating based on the above two targets (Fig. 1(b)), an η_{T+1_ave} and uniformity of 11.96% and 74.61% would be a good result at H of 150 nm and DC of 65% (marked by the black pentagram) respectively. Figure 1(c) shows the distribution of η_T+1 for this grating parameter as a function of angle, which gradually decreases as the FOV_X varies from −10 to 10°. Under a given FOV_X, the change in FOV_Y is less significant on η_T+1. Again, the optimized η_{T+1_ave} is apparently lower than that of asymmetric gratings (generally above 80%) [23,26], but the advantages in the nanoimprinting process make it still a competitive solution in the diffractive waveguide.

Because the diffracted light entering the waveguide has a high probability of interacting with the in-coupling DOE again, one need to consider the effect of BCL. As a demonstration, four incident beams (wavelength: 532 nm, spot size: 100×100 µm²) at two different positions are selected, with the light source 4.5 mm away from the in-coupling DOE (Figs. 2(a)–(d)). Two representative beam positions are used with the center of the projection at a distance of 0.8 and 4.2 mm from the left edge of the in-coupling grating. Two different angles of (−10°, 0°) and (10°, 0°) are assigned to beams #1, #2 and beams #3, #4. Since the T0 and T-1 (propagating along the negative x-axis direction) diffraction light cannot reach the out-coupling DOE, only the diffraction light at T+1 order that propagating along the positive x-axis is considered. The position and angle settings of the light sources described above include the two extreme cases of experiencing the shortest and longest light paths when reaching the out-coupling DOE. As seen in Fig. 2(a), the beam #1 will experience four times interaction with the in-coupling DOE in the waveguide and would cause severe BCL. Beam #4 no longer interacts with the in-coupling DOE after entering the waveguide, and thus has no BCL.

 

Fig. 2. Ray tracing diagrams of the two representative incident angles of (−10°, 0°) and (10°, 0°) are assigned to beams (a) #1, (b) #2 and beams (c) #3, (d) #4.

Download Full Size | PDF

The corresponding η_T+1, BCL and OEW of these four beams are listed in Table 1, where the OEW is defined as P_out/P_in, which is also numerically equal to “η_T+1-BCL-other diffraction losses”. The η_T+1 under (−10°, 0°) incident angle (beams #1 and #2) is apparently greater than that of the (10°, 0°) (beams #3 and #4). However, beams #1 and #2 have relatively severe BCL, with values of 3.59% and 1.97% respectively. Excluding the BCL and other losses, the detected OEWs of the four incident rays are 0.75%, 6.80%, 4.48% and 10.04%, respectively. These results suggest that the design of coupled gratings should not only focus on η_T+1, but also combine the BCL factors to obtain higher OEW.

Table 1. Corresponding η_T+1, BCL and OEW values for the four beams.

View Table

The design of in-coupling DOE, in the diffractive waveguide system, is aimed to maximize the power of light arriving at the out-coupling DOE, whose performance is quantized by the OEW. When the OEW is used as the optimization criterion, the area size of the light source is set to a realistic size of 3.4×4 mm² to ensure that the BCL effects of all incident light are taken into account. The obtained average OEW (OEW_ave) and its uniformity (Figs. 3(a) and 3(b)) associated with H and DC are quite different with the η_{T+1_ave} and its uniformity delivered by gratings only (See Supplement 1, Fig. S3). The high OEW_ave (Fig. 3(a)) occur in the lower left regions where H and DC are less than 400 nm and 55%, respectively. However, there is no clear regularity in the uniformity distribution in Fig. 3(b).

 

Fig. 3. The (a) OEW_ave and (b) its uniformity results as a function of H and DC, (c) FoM_OEW as a function of grating geometries and (d) the distribution of OEW with angle at H of 100 nm and DC of 40%.

Download Full Size | PDF

The optimized H and DC determined by the FoM algorithm (the product of the normalized OEWave and uniformity) are 100 nm and 40%, respectively (Fig. 3(c)), at which the OEWave and uniformity at this condition are 8.02% and 24.83%, respectively. The corresponding angle-dependent OEW under the optimized parameters is also shown in Fig. 3(d). This provides information on the intensity of reaching the out-coupling DOE and is more helpful for the design of the AR system. At the FOVX of −5°, the values of OEW are generally smaller than the other angles. Improving the OEWs in these angles are therefore an effective way to improve the OEWave and its uniformity.

Conformal deposition of a thin film with a higher refractive index (e.g., TiO₂, n = 2.33@532 nm) on the grating surface is a common strategy to improve diffraction efficiency [26]. It is also used in this work as an attempt to confirm whether it helps with OEW improvements. Using optimized parameters of 100 nm and 40% for H and DC respectively (See Supplement 1, Fig. S4a), the FOV dependent η_{T+1_ave} is calculated as shown in Supplement 1, Figure S4b. The strength of the efficiency has a similar distribution pattern to Fig. 1(b), with the η_{T+1_ave} and its uniformity of 11.4% and 55.78% respectively. A 20 nm TiO₂ conformal coated on the grating surface (Supplement 1, Fig. S4c) increases the η_{T+1_ave} to 13.4% and with a significantly improved uniformity up to 95.8% (Supplement 1, Fig. S4d). However, the OEW_ave and its uniformity of 7.82% and 9.31% calculated from the angle-dependent OEW (Supplement 1, Fig. S4e) are both lower than that of the structure without coating. The reason for this phenomenon is mainly due to the reversibility of grating diffraction. For example, a beam of light entering a waveguide from the air has a η_T+1 of 20%, another beam of light entering the air from the waveguide along the opposite direction of T+1 also has the same diffraction efficiency of 20%. In other words, the conformal coating increases the η_T+1 but simultaneously the BCL. The improvement of OEWs at FOV_X of 5° and 10° is attributed to the less interaction with in-coupling grating as illustrated in Figs. 2(c) and (d). These results indicate that full-area conformal coating on the grating surface can hardly improve the OEW_ave and its uniformity.

Region-selective coating depending on the incident angle may has the opportunity to improve the OEW at FOV_X of −5° by increasing the η_T+1 or suppressing the BCL effect. Conformal coatings on gratings can be rationally achieved by atomic layer deposition [33–35], but their nondirectional characteristic makes the region-selective deposition challenge. Here, we are attempting to improve the η_T+1 by depositing an interlayer with different refractive index in specific regions between the waveguide and the in-coupling grating as shown in Fig. 4(a). This process is prior to the formation of in-coupling gratings that can be realized by nanoimprinting lithography. Since interlayers are deposited on planar waveguides, deposition methods with high directionality such as sputtering and evaporation can be used by conventional hard masks.

 

Fig. 4. (a) Schematic diagram of the optical interlayer between the grating and the glass substrate. (b) The optimized η_T+1 obtained at two incident angles (−5°, 0°) and (10°, 0°) related to the interlayer materials. The angle-dependent η_T+1 with MgF₂/residual layers thicknesses of (c) 150 nm/700 nm, and (d) 200 nm/150 nm.

Download Full Size | PDF

A series of common optical materials such as MgF₂, SiO₂, Al₂O₃, HfO₂, ZrO₂ and TiO₂ are adopted as the interlayers to see their effects on the diffraction efficiency. It is important to note that the introduction of optical interlayer films should be accompanied by consideration of the effect of the actual processing on the grating. In the case of nanoimprinting processes, for example, there is usually an imprinted residual resin at the bottom of the grating, the thickness of which needs to be reflected in the modelling [36–38]. By sweeping the thickness of each optical interlayer and residual resin with respect to the two incident angles of (−5°, 0°) and (10°, 0°), the optimized η_T+1 and corresponding layer thickness are shown in Fig. 4(b) and Supplement 1, Table S1. As an example, the optimization process of the MgF₂ interlayer at these two incident angles is shown in Supplement 1, Figs. S5a and S5b. Typically, a higher n difference leads to a higher η_T+1 especially for the materials with a small n. For example, the highest η_T+1 of 45.83% and 51.44% are obtained for MgF₂ interlayer at both angles, while the Al₂O₃ interlayer with a small n difference only obtains a small η_T+1 improvement. In addition, we found that the method of the interlayer modulated diffraction efficiency features a high angular selectivity for a specific FOV range as shown in Figs. 4(c) and 4(d). The angular selectivity together with region-selective interlayer could benefit high η_T+1 at specific angles and suppress BCL at other angles. This unique light modulation behavior of the interlayer will bring a wide scope for improving the OEW and its uniformity.

As mentioned in Fig. 3(d), the low values in OEW pattern are generally located at FOV_X of −5°. On closer inspection of the relative positions of the light source and the in-coupling region in the optical waveguide system (Fig. 5(a)), the region I only interacts with light at negative angles of incidence. The thin-film interlayer is then inserted in Region I in an attempt to improve the OEW_ave and its uniformity. Similarly, the light sources with 5×8 directions are used where the thicknesses of the interlayer and residual resin vary in the range of 100∼500 nm and 100∼1000 nm. The MgF₂ with an n of 1.38 is employed as the interlayer, which obtain the maximum η_T+1 improvement at the incident angle of (−5°, 0°) as shown in Supplement 1,, Table S1. The optimized OEW_ave and its uniformity (Figs. 5(b) and (c)) that obtained from FoM_OEW criterion (Fig. 5(d)), are increased from 8.02% and 24.83% (Figs. 3(a) and (b)) to 8.34% and 35.02% for the interlayer and residual resin thicknesses of 200 nm and 700 nm, which are 4% and 41% higher than that without MgF₂ interlayer. It is worth noting that the optimized interlayer thickness here differs from the one of the gratings alone (Fig. 4(b) and Table S1) because the OEW takes into account the BCL. The angle-dependent OEW is plotted in Fig. 5(e) with a significant improvement at FOV_X of −5°, specifically from 2.02% to 3.9% at (−5°, ±7.5°) incident directions. And as expected, one cannot find any change with FOV_X of 0 and positive angles. The above studies demonstrate that region-selective interlayer could modulate the η_T+1 at specific angles of incidence and distribution of OEW, which combined with the diffraction characteristics of out-coupling DOE could improve the performance of AR systems.

 

Fig. 5. (a) Schematic diagram of the deposited thin-film interlayer in Region I. The (b) average efficiency, (c) uniformity and (d) FoM_OEW results as a function of thicknesses of thin-film interlayer and residual resin, where the optimized thicknesses of 200 nm (thin-film interlayer) and 700 nm (residual resin) are obtained by FoM algorithm as marked by the black pentagram in (d). (e) The distribution of OEW with angle at the thicknesses of 200 nm and 700 nm for interlayer and residual resin, respectively.

Download Full Size | PDF

To further investigate the effect of the thin-film interlayer on the efficiency and propagation pattern of diffracted light, the electric field (E) distributions of the gratings are calculated using the RCWA method [30]. The grating period, DC and H are set to be 390 nm, 40% and 100 nm, following the optimized parameters in Fig. 3(c). The MgF₂ thin-film (200 nm) is used as an interlayer upon which is the residual resin with a thickness of 700 nm following the FoM_OEW criterion as shown in Fig. 5(d). The thickness of the residual resin for the model without interlayer is set to 900 nm (Fig. 6(a)), which is equal to the total thickness of the MgF₂ interlayer and residual resin as indicated in Fig. 6(b). The wavelength and angle of incident light are set to be 532 nm and (−5°, 0°). Figures 6(a) and 6(b) show the electric field intensity ($|E |$) distribution without and with the MgF₂ interlayer, respectively. With the presence of MgF₂ interlayer (Fig. 6(b)), the enhanced spots like pattern resulting from the hybridization of cavity mode (along the z-axis) and Bloch mode (along the x-axis) [39,40] can be clearly seen. The cavity mode formed along the z-axis direction are originated from the refractive index variation between the grating, interlayer and residual resin, while the Bloch mode formed along the x-axis direction are generated due to the periodical property of the grating. In addition, from the perspective of the interference principle, the oblique diffracted light will not interfere with its own reflected light in the interlayer, but will interfere with diffracted light from the same direction at other positions of the grating. Therefore, we speculate that this interlayer modulation effect is likely to be caused by multi-beam interference.

 

Fig. 6. $|E |$ distributions of in-coupling DOEs (a) without and (b) with a MgF₂ interlayer, in which the wavelength and angle of incident light are set to be 532 nm and (−5°, 0°). (c) The η at T-1, T0 and T+1 orders as a function of diffraction angle for the gratings without and with a MgF₂ interlayer.

Download Full Size | PDF

The near-field simulation is difficult to demonstrate that the configuration with MgF₂ interlayer (Fig. 6(b)) holds higher η_T+1 because the electric fields of diffractions at T0, T-1 and T+1 orders are overlapped. Here, the η of these three diffraction orders with corresponding diffraction angles for both gratings are calculated and plotted in Fig. 6(c). Compared to the grating without interlayer, the diffraction angle of each order does not change, but the η_T+1 is increased from 14.23% to 37.28%, while the η_T-1 and η_T0 are decrease from 8.85% and 74.24% to 0.95% and 56.54%. These results reveal that the hybridized optical mode induced by interlayer could benefit the η_T+1 and suppress that of other orders. Meanwhile, the invariance of diffraction angle and higher η_T+1 make this novel grating suitable for using as in-coupling DOE, which could provide efficient coupling and high-quality image transfer in AR systems.

4. Conclusions

This work investigated the BCL effect that exists in the in-coupling DOE region of optical waveguide systems. The BCL reduced the light source utilization, especially at negative angles of incidence. A more accurate method using OEW criterion was proposed for assessing the performance of in-coupling DOE. After optimizing the H and DC of the grating, the OEW_ave of 8.02% with a uniformity of 24.83% was achieved. Thin-film interlayers, inserted between the grating and the waveguide, were demonstrated to enhance the η_T+1. The angular selectivity of the method together with region-selective interlayer could benefit high η_T+1 at specific angles and further enhance the OEW. The optimized OEW_ave and its uniformity were increased to 8.34% and 35.02% for the grating with selectively inserted MgF₂ interlayer, which are 4% and 41% higher than that without MgF₂ interlayer. In addition, the introduction of interlayer allows the η of each order to be redistribution and doesn’t affect the propagation angles of the diffracted light. The interlayer scheme provides a simple and effective method for angularly selective modulation of diffraction efficiency and can be applied in the design of fold or out-coupling DOEs that are subsequently applied to AR systems.