1. Introduction

Resonant waveguide gratings (RWGs) are dielectric grating structures that support leaky guided modes strongly resonant within the light cone, also known as guided mode resonances. In the past few decades, these structures have seen applications in a variety of fields, including spectroscopy, signal processing, sensing devices and optical communications [1]. Different from common guided and quasi-guided mode structures, such as Bragg gratings and 2D photonic crystal (PhC) slabs, RWGs have the advantage of being compact and relatively simple to fabricate. RWGs have been traditionally explored for their use as reflection filters in the classical mounting configuration (Fig. 1), but they can also be used as transmission filters, and have been explored for conical mounting [2]. Their design can be aided by the use of Effective Medium Theory (EMT), which has been developed at arbitrary solid angle of incidence for both polarizations [3,4]. Much work has been done to tune the angular [5] and polarization response of these structures (for instance, achieving either polarization-dependence [6] or polarization-independence [7]), and to reduce the reflection sidebands [8].

 

Fig. 1. (a) Schematic geometry showing how a heads-up display projects onto a semi-transparent windshield to create a virtual image. (b) Schematic geometry of the RWG and the excitation angles. The grating is embedded in a thick glass matrix. (c) RWG geometry. The azimuthal angle α determines the plane of incidence. Classical and fully conical mounting correspond to α = 0° and α = 90° respectively. The incident light is TM polarized when the electric field is in the plane of incidence and TE polarized when perpendicular to the plane of incidence.

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In this work, we investigate the use of RWGs to obtain high reflection at selected wavelengths and polarization for oblique incidence, while maintaining large transmission for unpolarized broadband excitation. Such functionality is ideally suited to implement projection combiners for automotive heads-up displays (HUDs), as schematically sketched in Fig. 1(a), which require high transmissivity to sunlight or ambient light and high reflectivity for the selected HUD image colors (e.g., blue, green and red wavelengths) within a small solid angle. The specular reflection of these colors (emitted by a laser or LED source at a select polarization) can be then used to create a desired virtual image to be superimposed to the background (Fig. 1(a)). There are several constraints to realize this functionality effectively: the resonance linewidth and angular bandwidth must be carefully tuned to maintain a large transparency, and at the same time enable efficient image projection. Normally, it is difficult to meet these requirements with RWG’s, due to their inherent dispersive nature: the resonance wavelength at a given solid angle excitation will typically shift by an amount comparable to or larger than the linewidth when the excitation angle changes by a few degrees. In particular, the angular-to-spectral linewidths in RWGs are about ɛ = 0.1–0.5° nm⁻¹, which limits the angular tolerance of these spectral resonances [9]. One approach to increase this tolerance is to use doubly periodic structures, which have been analyzed for both gratings and PhC slabs [9,10], but which strictly operate near normal incidence. In this study we show that it is instead possible to realize a single-layer RWG yielding three-color selectivity while still maintaining good angular dispersion.

We consider a RWG embedded inside a uniform glass-like matrix (Fig. 1(b)). For simplicity, we assume that the embedding medium has refractive index n = 1.5, which we will refer to as ‘glass’. For the RWG, we consider titanium dioxide (TiO₂) due to its relatively large index and low loss in the visible spectrum. The index of our sample was experimentally found to range from 2.6 to 2.3 between wavelengths of 400 and 700 nm. The operational excitation angle, measured with respect to the normal to the grating plane, is set to $\theta =$ 58° when measured in air (Figs. 1(b-c)). Due to refraction, this excitation angle corresponds to ${\theta _{glass}} = \; 34.4^{\circ} $ in glass (see also schematic in Fig. 1(b)). In the following, we always refer to angles measured in air. The particular angle $\theta =$ 58° is chosen because (i) it is large enough to accommodate a HUD projector and to account for the typical position of the user in automotive applications; and importantly, (ii) under p-polarized excitation it approximately matches the Brewster angle at the two air/glasses interfaces, for which no reflection occurs. The desired angular linewidths over which high reflection must arise (at the desired wavelengths) is set to $\mathrm{\Delta }\theta = \mathrm{\Delta }\alpha \; $=6° for both polar and azimuthal angles, in order to match the typical field of view of standard HUDs projectors. The grating designs (Fig. 1(c)) are characterized by the period ${\bigwedge}$, fill factor f, grating thickness d_g and waveguide thickness d_w.

A typical challenge in using periodic gratings for virtual reality displays is the undesired scattering of light into higher diffraction orders, which may give rise to artifacts. In recent efforts, these unwanted features were suppressed by using thin grating structures designed to deflect infrared light for the purpose of eye tracking [11], at the expense of coupling efficiency. In this work, we mitigate these artifacts by using subwavelength periods and exploiting the total internal reflection at the air/glass interface to trap most of the light scattered in the first diffraction order. Specifically, for a period of 250 nm, i.e., the largest period considered in the following, the 1^st diffraction order for any wavelength above ∼465 nm will be at angles (measured in glass) larger than ∼42°, for which total internal reflection occurs, thus trapping light in the glass layer. Therefore, only a small portion of the visible spectrum can escape the glass and potentially contribute to rainbow artifacts.

2. Design principles and optimization

TM (or p-polarized) Operation. The design goals of the target device, i.e., high reflectance and angular tolerance at three specific wavelengths, and yet maximal transmittance outside these wavelengths, are ambitious. To tackle this challenge, we employ well-established computational techniques to widely search the design space and make optimal use of the degrees of freedom at our disposal. In particular, we use inverse-design, or adjoint optimization [12–14], which is based on a gradient descent approach and whose main details are quickly summarized here below. Interestingly, despite the fact that the optimization procedure is free to generate any arbitrary two-dimensional unit cell, the resulting device geometry always converged to that of a 1D grating, suggesting that the RWG is a natural choice for these applications. Within our implementation of the inverse-design method, the structure is defined by a spatially varying permittivity distribution within a rectangular design region. The grating layer thickness is fixed to d_g = 155 nm and, for simplicity, we assume a dispersion-less refractive index for both TiO₂ ($n = 2.4$) and the embedding low-index material (${n_{low}} = 1.5)$. These materials are compatible with large-area fabrication of the proposed devices. The periods along the y and x directions are set to 150 and 360 nm, respectively. The target multifunctional response of the device is defined in the form of an objective function given by

Figure 2(a) shows the structural evolution of the in-plane unit cell for different iterations. It is observed that the permittivity profiles, which initially start from a random distribution, gradually evolve into a 1D grating consisting of a binary permittivity distribution. The corresponding reflectance for the three wavelengths over the course of the iteration process is shown in Fig. 2(b), from which we see that the overall reflections gradually increase for the red and blue wavelengths. While no improvement on the objective function is observed for green light, the reflection spectrum of the optimized device reveals three reflection peaks at the wavelength of 450, 560 and 600 nm (Fig. 2(c)), corresponding roughly to the RGB wavelengths. As is indicated by the linewidths, the polar tolerance at 560 nm is expected to be worse than at 450 and 650 nm, which is confirmed experimentally in the following. It is remarkable to see that a single grating device, suitably optimized, can support three distinct narrowband peaks aligned with the target wavelengths, operating as a convenient platform for the application of interest. The parameters of this optimized design are ${\bigwedge}$ = 360 nm, f = 0.819, d_g= 155 nm and d_w= 0.

 

Fig. 2. RGB reflector optimization. (a) Structural evolution during the optimization process. The permittivity profiles are selected at the iteration steps of 1, 75 and 150. (b) Reflections at RGB (450, 530 and 600 nm) over the course of iteration. (c) Reflection spectrum of the optimized device, showing three reflection peaks at the wavelengths of 450, 560 and 600 nm.

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TE (or s-polarized) Operation. As mentioned above, using TM polarization for the incident signal has the important advantage that, close to the Brewster angle, the reflection at each air-glass interface is minimal. When considering TE polarization (i.e., s-polarized) at the same angle, each glass-air interface will reflect about 16% of the input intensity, thus making TE polarization less attractive for our proposed approach for a HUD system. Nonetheless, we explored the possibility of using TE polarization to increase the design flexibility, and for other potential applications where the absolute reflection level is not as critical or in situations in which anti-reflection coatings may mitigate these issues.

Here, we have optimized single-resonance TE gratings in the classical mounting with no residual layer (i.e. d_w= 0) to support single-color operation, with the following parameters: ${\bigwedge}$ = 180 nm, f = 0.5, d_g = 50 nm for blue, ${\bigwedge}$ = 220 nm, f = 0.32, d_g = 90 nm for green and ${\bigwedge}$ = 255 nm, f = 0.24, d_g = 120 nm for red. Each of these devices supports a resonance with robust angular response between 55° and 61° for the azimuthal angle (Fig. 3(a), top). We found that these devices are also robust against angular variations in the azimuthal direction. Under TM- polarized excitation, the grating supports resonances at approximately the same wavelengths but with narrower bandwidths (Fig. 3(a), bottom). We tried to design a single grating supporting three resonance peaks, similar to the design shown in Fig. 2. In this case, however, it was more challenging to control the resonant wavelength positions and their angular tolerances, since the TE polarization is characterized by a purely tangential electric field for any incidence angle, limiting the forms of resonant interactions with the grating fins.

 

Fig. 3. Cascaded BG-filter and R-filter. Simulations assume glass as the background medium. (a) 0^th order TE (top) and TM (bottom) reflectance at 55° and 61° for three RWG filters supporting blue, green and red resonances as colored. (b) 0^th order TE (top) and TM (bottom) reflectance at 58° of the BG-filter (blue line) and R-filter (red line). (c) Cascaded design geometry. (d) Reflection level as a function of the separation between the filters near the wavelengths of 452 nm, 536 nm and 611 nm. (e) Top: 0^th order TE reflectance spectra of the cascaded devices for $\theta = 58^\circ $. Bottom: Fraction of energy scattered into higher diffraction orders (in glass).

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We therefore considered a different approach to realize a three-color device, in which two gratings, one supporting a dual blue and green resonance (termed BG-filter) and a second one supporting a red resonance (termed R-filter) are stacked on top of each other, as sketched in Fig. 3(c). In Fig. 3(b) the 0^th order TE and TM reflection spectra at 58° are plotted for both devices, with parameters ${\bigwedge}$ = 190 nm, f = 0.37, d_g = d_w = 85 nm for the BG-filter. When considering the two gratings separately, the minimum 0^th-order TE reflections between 55° and 61° [denoted as R_m(λ)] at λ = 452 nm, 536 nm (BG-filter) and 611 nm (R-filter) are 0.52, 0.35 and 0.35 respectively. When the elements are cascaded, the top of the resonance bands can flatten and angular tolerance can be improved, a principle that has been applied to cascaded identical gratings in recent papers [16,17]. We assume here a sufficiently large spacing d_s so that evanescent coupling is negligible, and the spectra are therefore independent of the in-plane displacement of the two gratings [18]. The numerical tool used for simulations (RCWA) requires the structure to have a fixed periodicity, while the R and BG filters have different pitches ${\bigwedge}$. To overcome this problem, we define a super-period that is a common multiple of the two periods and repeat unit cells accordingly. Here the chosen super-period is ${\bigwedge}$_super= 765 nm.

When cascaded, Fabry-Perot interference is observed, and as d_s increases the three wavelengths for which R_m is greatest are slightly perturbed. Figure 3(d) shows R_m at these wavelengths as a function of d_s, from which we identify d_s ≈ 300 nm as the optimal spacing. While R_m decreases to 0.47 at the blue wavelength, it also increases to 0.44 and 0.5 at green and red. This scenario, considered in Fig. 3(d), displays a greater interference amplitude for the red resonance than for the green, which is understood by the fact that the reflectance tail of the BG-filter is higher at red wavelengths than the reflectance tail of the R-filter is at green wavelengths. The spectrum of the cascaded device is shown in Fig. 3(e) along with the power leakage into higher diffraction orders (bottom plot), which originates primarily from the BG-filter and the R-filter separately than from the cascaded configuration. It is important to note that this analysis has been performed in glass as in the TM case for comparison. Taking to account the reflections at the air/glass interfaces, we found negligible power leaking into higher diffraction order past ∼465nm as expected. These interfaces also enhance the broadband reflection in Fig. 3(e). For our analysis we found good agreement with experimental results (for incident angles up to 60°) by applying Fresnel coefficients, although other methods exist to more rigorously take into account incoherent light propagation in the glass [19].

3. Experimental results

Based on these designs, we fabricated 1D titania gratings immersed into an n≈1.5 embedding layer for TM and TE operation using standard electron-beam-lithography (EBL) and R2R fabrication methods. The angle-dependent transmission and reflection spectra were acquired with custom-built setups (see Supplementary Information for details on fabrication and measurements). By using collection/excitation lenses with long focal lengths (f = 10cm), we collect only the 0^th transmitted/reflected order. Thus, the measured transmission/reflection spectra allow us to set an upper bound to the light intensity scattered into additional diffraction orders.

We started by fabricating and characterizing a grating for three-color TM operation, with a geometry consistent with the one described in Fig. 2, via EBL. The sample transmission versus wavelength and incident angle for TM input polarization (as sketched in Fig. 4(a)) is shown in Fig. 4(b). For small angles, the spectrum is dominated by a single resonance at $\lambda \approx 640\; nm$. At shorter wavelengths ($\lambda < 550\; nm)$, the transmission is lowered due to the onset of the first diffraction order. As the angle increases, two additional modes appear, whose excitation at $\theta = {0^o}$ is not possible due to symmetry. Around the angle of operation ($\theta = {58^o}$) the dips of the three modes align well with the desired wavelengths (Figs. 4(e)-(f)), in agreement with Fig. 2(c). The transmission spectra for TE polarization (Fig. 4(c)) also show three dips, although with narrower linewidths. As mentioned above, an important figure of merit to ensure that these devices can be used for augmented reality applications is that, apart from the wavelengths of operation, they remain highly transmissive for a broadband signal of arbitrary polarization in the visible range. To verify this feature, we calculated the average between the two datasets in Figs. 4(b)-(c), emulating an unpolarized beam, and we further averaged across all wavelengths in the visible range. The obtained curve (Fig. 4(d), blue curve) shows that the average unpolarized transmission is about 75% at normal incidence, dropping to about 65% at the angle of operation $\theta = {58^o}$. This slow decrease in average transmission as $\theta $ increases is mainly attributed to the reduced transmission of the TE component at large angles, due to the large reflection at the air/glass interfaces at large angles. We anticipate that these metrics of performance are well suited for augmented reality applications. For comparison, the red curve in Fig. 4(d) shows the average unpolarized transmission for a bare thick glass slab.

 

Fig. 4. Experimental characterization of TM-based RGB gratings. (a) Schematic of the sample and excitation geometry. For TM polarization, the magnetic field lies in the sample plane, and it is orthogonal to grating wires. (b-c) Color-coded transmission spectra of the grating versus angle $\theta $ and wavelength for TM (b) and TE (c) incident polarization. (d) Unpolarized averaged transmission versus $\theta $, obtained by averaging the maps in panels (b) and (c) and further averaging spectrally. (e) TM transmission spectra at three selected angles, as described in the text. (f) Zoom-in of panel (e). (g) Reflection (blue curve) and transmission (red curve) spectra of the grating at $\theta = {58^o}$. The thick yellow line shows the sum of the two curves.

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In order to confirm that the transmission dips are due to large reflections, and to quantify the overall absorption, we measured reflection spectra at selected angles (see Supplementary Information). In Fig. 4(g) we compare the reflection spectra at $\theta = {58^o}$ (blue line) with the transmission spectra (red line) at the same angle. Due to limitations in our laser source, it was not possible to measure reflection at wavelengths shorter than 500 nm. The transmission dip at $560\; nm$ and $\lambda \approx 600\; nm$ are accompanied by corresponding reflection peaks with almost equal magnitude. The sum of the transmission and reflection spectra (“R + T”, thick yellow line in Fig. 4(g)) is unitary off-resonance. At resonance, the R + T drops by about 15–20%, indicating the emergence of resonant loss. This can be attributed to fabrication imperfections and to some residual of the metallic mask used for fabrication.

We also fabricated a TE-based R-filter consistent with our simulations in Fig. 3, with experimental results shown in Fig. 5 for the excitation configuration sketched in Fig. 5(a). The average unpolarized transmission (Fig. 5(b)) is about 80% at normal incidence and drops off faster than the transmission of the slab at large angles. At $\theta = {58^o}$ the reflection/transmission resonance (Fig. 5(c)) is located near 600 nm, as predicted, and the level of loss is comparable to the one seen in Fig. 4(g). At all angles, the spectrum is found to only support a single TE and TM resonance, with the TM resonance being much narrower. We note that the reflection spectra (blue curve in Fig. 5(c)) feature a nonresonant reflection level of about 25%. This large background reflection is due to the two air-glass interfaces, each of them reflecting about 16% of the TE polarized light at this angle. This issue can be minimized by adding anti-reflection coatings on the two interfaces. The response of the TE BG-filter (Fig. 6) is also shown to match well with our simulated response in Fig. 3. The sub-optimal transmission dip for the green resonance (Fig. 6(c)) is attributed to the surface roughness of the sample, and it can be understood as resulting from Rayleigh scattering which, given its ${\lambda ^4}$ dependence, has a much larger impact for shorter wavelengths.

 

Fig. 5. Experimental characterization of the TE-based R-filter. (a) Schematic of the sample and the excitation geometry. For TE polarization, the electric field lies in the sample plane, and it is parallel to the grating wires. (b) Unpolarized averaged transmission versus $\theta $, obtained by averaging the maps in panels (d) and (e) and further averaging spectrally. (c) Reflection (blue curve) and transmission (red curve) spectra of the grating at $\theta = {58^o}$. The thick yellow line shows the sum of the two curves. (d,e) Color-coded transmission spectra of the grating versus angle $\theta $ and wavelength for TE (d) and TM (e) incident polarization.

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Fig. 6. Experimental characterization of the TE-based BG-filter. (a,b,d,e) Same as in Fig. 5. (c) TE Transmission spectra at three selected angles.

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While the experimental results discussed in Fig. 4–6 match quite closely the simulated one, small deviations are also present, which can be attributed to expected tolerances in the EBL fabrication process. The design of each 1D grating is fully identified by the four parameters ${\bigwedge}$, f, d_g, d_w. The spectral position of the transmission dips is mainly controlled by the lattice constant Λ. The accuracy of Λ in the fabricated devices depends only on the resolution of the EBL writing process, which is typically better than 2 nm. This allows us to achieve an excellent agreement between the spectral positions of the measured transmission dips in Fig. 4(e) and the simulated ones in Fig. 2(c), with a maximum deviation of less than 3 nm for the green and red dips, and ∼10nm for the blue dip. The other parameters (f, d_g, d_w), while still playing a minor role in the spectral position, have typically a bigger impact on the spectral lineshapes. In fabricated devices, the width of the ridges (f${\bigwedge}$) is typically larger (by ∼10–20 nm) than the designed one, due to proximity effects in the electron dose. This can be partially corrected by running a first calibration test, and reducing accordingly the designed value of f. The accuracy of the thicknesses d_g and d_w is instead determined by the precision and reproducibility of the etching processes. In our experiment, we typically obtained a maximum deviation of 5 nm from the targeted thicknesses.

As demonstrated by the proof-of-principle experiments in Figs. 4–6, these devices are promising for augmented reality applications; however, the fabrication technique considered so far (EBL) is limited to device footprints of few millimeters, and it cannot be used to fabricate large-area displays for practical uses in automotive applications. To demonstrate that these designs can be implemented in a realistic setting, we also investigated the use of roll-to-roll (R2R) techniques, which are suitable for industrial fabrication over large scales, and we explored their implementation and characterization. In this approach, a patterned polymer layer is coated by a high-index grating material (TiO₂ in our case), and then a second polymer layer is used to cap the structure. Figure 7(a) shows an example of a fabricated device prior to the deposition of the second polymer layer. Because the TiO₂ deposition is conformal, a base layer (corresponding to half of the waveguide thickness d_w in Fig. 1) is generated as a result of this fabrication approach, although not ideal for some designs (e.g., for the TE-based R filter in Fig. 4(a)). We can however take its presence into account and correct for it by slightly perturbing the other parameters. In Fig. 7, for example, we consider a design optimized for TE reflection at red wavelengths, with ${\bigwedge}$ = 235 nm, f = 0.4, d_g = 105 nm, d_w =47.5 nm. We fabricated (Figs. 7(a)-(b)) a large-area metasurface (∼3.5 inches in linear dimension) of this design with the R2R technique, which consisted in the following steps: a 100mm wafer was patterned using Talbot lithography, creating a 94mm diameter area with the 235nm pitch TE-based R-filter grating pattern. A polymer copy of the patterned wafer was created using a UV-curable resist, and then the polymer copy was replicated into a metal sleeve. The sleeve with the grating pattern was installed into a custom R2R nanoimprint lithography system where a UV curable resist was coated onto a polymer film and placed in contact with the pattern, cured and peeled from the tool. The cured UV resist has a refractive index approximately n = 1.5. In this fabrication run, approximately 320 copies of the grating were created on film in a time span of about 20 minutes, highlighting the high throughput capabilities of this method. The nanopatterned film was then cut from the roll and coated with 70nm of TiO2.

 

Fig. 7. Experimental results of large-area TE R-filter fabricated with roll-to-roll techniques. (a) SEM image of a cleaved sample, dark areas are made of TiO₂ and lighter areas are the glass-like patterned polymer (scalebar = 500 nm). (b) Picture of the large-area sample. (c) Angle-resolved transmission spectra under TE excitation (see Fig. 5(a)). (d-e) TE transmission (d) and reflection (e) spectra at three selected angles.

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The fabricated sample was characterized optically with the same setup used for the EBL samples The angular-resolved transmission spectra (Fig. 7(c)) show the expected resonance, similar to the device fabricated via electron-beam-lithography (Fig. 5(d)). The periodic transmission modulations observed in Fig. 7(c) are due to the fact that, in these samples, the embedding polymer layer was kept quite thin (<300 µm), thus leading to non-negligible Fabry-Perot features. The transmission and reflection spectra at selected angles [Figs. 7(d-e)] shows that the sum of reflection and transmission is above 90% on resonance, demonstrating that the R2R devices feature much less insertion loss than the EBL ones. On the other hand, the nonresonant reflection level (∼40%) in Fig. 7(e) is substantially higher than the one observed in the corresponding EBL sample (∼25%) in Fig. 5. As clear from Fig. 7(a), the waveguide layer (i.e., the layer of thickness d_w in Fig. 1(c)) is not flat, but it is characterized by periodic grooves originating from the conformal deposition process. Since the thickness d_w of this layer has quite a large impact on the overall nonresonant reflection, we attribute the large background reflection observed in Fig. 7(e) to these unwanted grooves. Further small deviations between the results in Fig. 7 and Fig. 5, in particular an overall spectral shift of about 20nm, are also likely due to the non-planar TIO₂ layer.

4. Conclusions

In this work, we explored the design, optimization, fabrication and characterization of RWGs for TM and TE operation for augmented reality applications. For TM operation, we simulated and fabricated a single RWG structure supporting three RGB resonances at oblique incidence, which meet the angular tolerances and spectral features required for these applications. To expand our study and to further demonstrate the capabilities of RWGs, we investigated numerically and experimentally also TE single-color and two-color devices, and we explored cascading these structures to realize a three-color operation. These devices also can meet the angular tolerance required for augmented reality. Comparing the two operations, we found that when a grating is optimized for TE response, its TM response is characterized by very narrow resonances, which makes it easier to increase the average unpolarized transmission. On the other hand, TE operation suffers from an unavoidable large reflection at the glass/air interfaces for large angles.

As indicated by the angle-dependent spectra of the fabricated structures, RWGs suffer from dispersion, which causes the resonances to shift to different wavelengths when changing angle. Consequently, the spectral linewidths must be fairly large (∼60nm) to achieve a typically desirable angular tolerance of 6°. Despite these limitations, we have shown that RWGs can be an attractive solution for multi-color HUD displays, due to their ease of design and fabrication, and compatibility with large-area fabrication methods.