## Abstract

Uniform illuminance over the expanded exit pupil is an important requirement for waveguide display systems with a wide field of view (FOV). To address this issue, we develop a monochromatic binocular waveguide display in this paper. Two surface-relief diffraction gratings are designed as in-couplers and out-couplers. The parameters of the gratings are optimized to achieve uniform diffraction efficiency distributions over a broad angular range. The grating couplers enable the system to realize a diagonal FOV of 40°. A freeform surface prism is designed as the projection optics. The diameters of the two exit pupils are 12 mm in the expanding direction at an eye relief of 19 mm.

© 2017 Optical Society of America

## 1. Introduction

With the rapid development of augmented reality (AR) technology, near-eye see-through displays are attracting considerable attention and have potential to be widely applied in navigation, military, education, and entertainment. Several large corporations, such as Google, Sony, and Microsoft, have also released commercial products.

AR devices can superimpose virtual images over the real-world scene and the user can observe both simultaneously. The exiting approaches include freeform optical prisms [1,2], projection systems [3,4], retina scanning [5], reflective systems [6], hybrid reflective-refractive systems [7], and optical waveguides [8,9]. Among those approaches, optical waveguide technology provides the most promising means to realize a light and compact structure. In addition, optical waveguides can expand the exit pupil along the direction of propagating light inside it. Generally, a waveguide display system consists of a micro-display, collimator, and waveguide optics, which integrates an in-coupler and out-coupler. According to different working principles of couplers, waveguide technology can be classified into geometrical waveguides and diffraction waveguides. For example, Lumus [10,11] employed a reflective mirror as an in-coupler and a partially reflective mirror array (PRMA) as an out-coupler. Optinvent [11] adopted the surface reflector approach as an out-coupling subsystem. Furthermore, based on diffraction theory, Microsoft [12,13], Nokia [14,15], Sony [16,17], and BAE Systems [18,19] have used diffraction or holographic gratings to develop their waveguide display systems.

For a waveguide display system, the field of view (FOV) is a key parameter for evaluating its optical performance. The angular range, in which the couplers need to have uniform reflectance or efficiencies, is the main factor limiting the FOV. There are various ways of widening the FOV. As the diffraction efficiency of volume holographic gratings (VHGs) is sensitive to the incident angle, Han et al. [20] proposed triple-multiplexing VHGs to achieve a broad angular bandwidth and theoretically realized a wide diagonal FOV of 45°. By optimizing the parameters of the PRMA, the FOV in the design by Wang et al. [21] reached 36° in the pupil-expanding direction. The slanted surface-relief grating proposed by Levola [14] can provide a FOV of 25° with a refractive index of 1.71 for the substrate. Another key parameter for ultra-thin waveguide display systems is the illuminance uniformity over the expanded exit pupil. As the propagating light inside the waveguide is repeatedly diffracted by the out-coupler, with a portion of the light exiting the waveguide each time, the luminance of the exit light decreases along the pupil-expanding direction. To address this issue, the common approach [22] involves raising the out-coupling efficiency gradually, by changing the depth of the grating. Travis [13] introduced an embedded partially reflective mirror (EPRM) which is parallel to the surface of the waveguide. The EPRM can turn a single exit light ray into multiple exit light rays to make the illuminance distribution more uniform. However, because of the divergence of light paths inside the waveguide, the out-coupler should have the appropriate reflectance or diffraction efficiencies for different angles in different positions along the light paths, which increases the designing difficulty and thus limits the FOV. Therefore, it is hard to realize uniform illuminance over the expanded exit pupil while maintain a wide FOV.

To address this issue, we propose a binocular waveguide display system with diffraction gratings and freeform optics. To realize a wide FOV, we design two surface-relief diffraction gratings as in-couplers and out-couplers. Based on rigorous coupled wave analysis (RCWA) [23,24], we optimize the parameters of the gratings to achieve uniform efficiency distributions over a broad angular range. The diagonal FOV of the binocular display reaches 40°. According to the symmetry structure of the grating out-coupler, we combine the two separated waveguides in the subsystems to improve the illuminance uniformity. Furthermore, a freeform surface (FFS) prism matching this combined waveguide is designed as the projection optics. The diameters of the two exit pupils are 12 mm in the expanding direction at an eye relief of 19 mm. Uniform illuminance over the expanded exit pupils and a wide FOV are realized according to the simulation using this approach.

## 2. Design of diffraction grating couplers

A schematic of a waveguide display configuration is illustrated in Fig. 1(a). Light from the micro-display is collimated and then coupled into the waveguide by the in-coupler. The out-coupler serves to radiate the image-bearing light out of the waveguide. Therefore, the couplers are the key elements determining the range of the FOV. When designing diffraction grating couplers, there are some conditions that need to be met. First, in the in-coupling diffraction, the energy of the in-coming light should concentrate on the first reflective order (${R}_{1}$). The diffraction angle of ${R}_{1}$ should meet the total internal reflection (TIR) condition. Thus, the light in ${R}_{1}$ can be restricted within waveguide and propagate to the out-coupler. Second, to avoid stray light, no diffractions higher than ${R}_{1}$ should exist in the in-coupling. Third, in the out-coupling diffraction, the energy of the propagating light should concentrate on the zeroth reflective order (${R}_{0}$) as well as the minus first reflective order (${R}_{-1}$). The light in ${R}_{0}$ keeps propagating inside waveguide and the light in ${R}_{-1}$ exits the waveguide and propagates to the observer. Fourth, to realize uniform illumination for different fields, the diffraction efficiencies of the two couplers should remain stable when the incident angle of the in-coming light varies in the range of the FOV.

To meet the conditions mentioned above, the parameters of grating couplers are discussed and determined as follows. In the diffraction geometry described in Fig. 1(b), the K-vector of the in-coming light is

*m*th diffracted order inside the waveguide is

*T*is the period of the grating in-coupler. The TIR condition for ${R}_{1}$ leads to an upper boundary for the period

*T*of the grating in-coupler:The requirement that no diffractions higher than ${R}_{1}$ exist in the in-coupling can be expressed by the following equation:

From Eqs. (4) and (5), we know that the determinations of the period *T* of the grating in-coupler and the refractive index ${n}_{1}$ of the waveguide material rely on the range of the FOV. In our design, the FOV is set as 33°(H) × 24°(V) and the wavelength $\lambda $ is 532 nm. To satisfy the equations above, the period *T* of the grating in-coupler is set as 380 nm. We choose the N-LAF33 glass as the waveguide material because of its high refractive index and relatively low density. The refractive index ${n}_{1}$ of the N-LAF33 glass is 1.792 at 532 nm. For the grating out-coupler, its period should equal that of the in-coupler to keep the angle of the exit light the same as that of the in-coming light, and to thus avoid distorting the image displayed. It can be further proven that the angular range of the minus second reflective order (${R}_{-2}$) in the out-coupling is the same as that of ${R}_{1}$ in the in-coupling. Therefore, the light in ${R}_{-2}$ is also restricted within the waveguide and it does not affect the projected image.

Figure 2 illustrates the grating structures we proposed and the diffraction order distributions. Silver is selected as the bottom material of the grating in-coupler. The complex refractive index of silver is about 0.13 + 3.25i at 532 nm, which is low in real part and high in imaginary part. It helps to reflect more transverse-electric (TE) component and absorb more transverse-magnetic (TM) component of the in-coming light. Therefore, the in-coupled light can be polarized to TE mode after optimization and we do not need to consider the polarization sensitivity of the grating out-coupler. A thin film of titanium dioxide (TiO2) is coated on the surface of the tooth-profiled glass. Because the refractive index of TiO2, which is about 2.98 at 532 nm, is much higher than the waveguide, it can make the diffraction efficiencies less sensitive to the incident angle. ${\theta}_{ih}$ and ${\theta}_{iv}$ denote the horizontal and vertical incident angles inside waveguide. For the grating in-coupler, ${\theta}_{ih}$ ranges from –9.0° to 9.0° and ${\theta}_{iv}$ ranges from –6.5° to 6.5°. For the grating out-coupler, ${\theta}_{ih}$ ranges from 38.5° to 70.5° and ${\theta}_{iv}$ ranges from –18.5° to 18.5°.

The diffraction efficiencies of the two gratings are calculated from the theory of RCWA by the software Gsolver. Based on the calculation results, the grating parameters including the two groove angles and film thickness, are optimized. In the optimization, the average value and uniformity across the efficiency distributions with the incident angle, have both been taken into consideration. The efficiency uniformity is defined as follows:

## 3. Uniform illumination of combined waveguide

Generally, a binocular waveguide display consists of two separated monocular subsystems. Because of the repeated diffractions of propagating light inside the waveguide, the illuminance over the expanded exit pupil decreases along the expanding direction. The symmetry structure of the grating out-coupler allows this issue to be addressed by combining the two separated waveguides in monocular subsystems. By this approach, the image-bearing light from the micro-display at either side of the combined waveguide can reach both exit pupils. Thus, comparing the separated waveguide with the combined one, the single input turns to a double input. Figure 4(a) shows the combined waveguide system. Collimated light in all fields is coupled into the waveguide at two sides and propagates to both exit pupils. The size of the combined waveguide is 129 mm × 50 mm × 1.6 mm with widths of 12 mm for the grating in-coupler and 25 mm for the grating out-coupler each side. At an eye relief of 19 mm, the exit pupils, $EP{D}_{Y}$ in the horizontal direction and $EP{D}_{X}$ in the vertical direction are 12 and 5 mm, respectively. The distance between the centers of the two exit pupils is 60 mm and the distance between the in-coupler and out-coupler is 10 mm.

Detailed ray paths of the light in the marginal field inside the combined waveguide are illustrated in Fig. 4(b). $\theta {\text{'}}_{1l}$, $\theta {\text{'}}_{-1l}$ and $\theta {\text{'}}_{1r}$, $\theta {\text{'}}_{-1r}$ refer to the diffraction angles of ${R}_{1}$ and ${R}_{-1}$ at the left and right sides, respectively. The energy of the in-coming light highly concentrates on ${R}_{1}$. It should be noted that the grating in-couplers at both sides are opposite each other such that $\theta {\text{'}}_{1r}$ equals $\theta {\text{'}}_{-1l}$ and $\theta {\text{'}}_{1l}$ equals $\theta {\text{'}}_{-1r}$. The relationship between $\theta {\text{'}}_{1l}$ and $\theta {\text{'}}_{1r}$ is described by

*Y*direction with respect to the norm of the waveguide surface, $\theta {\text{'}}_{1l}$ will become smaller than $\theta {\text{'}}_{1r}$, and vice versa. A smaller diffraction angle leads to more TIRs inside the waveguide and a more rapid decrease in luminance, correspondingly. Accordingly, for the exit light in the margin fields (–16.5°H, $\pm $12°V) from the left side, its luminance decreases the fastest, while for that from the right side, its luminance decreases the slowest along the opposite direction.

To investigate the illuminance distributions over the expanded exit pupil, ray-tracing simulations are implemented by the software TracePro. The in-couplers and out-couplers integrated into the waveguide are the diffraction gratings designed in Section 2. As the combined waveguide is symmetrical, we take the left exit pupil as an example to illustrate the illuminance distributions. Figures 5(a), 5(b), and 5(c) show the ray-tracings of the exit light in all fields that reaches the left exit pupil when the in-coming light with the same luminance is input at the left, right, and both sides of the combined waveguide, respectively. The corresponding normalized illuminance distributions of the marginal field (–16.5°H, 12°V) are presented in Figs. 5(d), 5(e), and 5(f). As we can see, the illuminance in Figs. 5(d) and 5(e) decreases along opposite directions. Therefore, the superposed distribution in Fig. 5(f) can be more uniform as the decrease is mutually compensated.

Similarly, we use the uniformity to evaluate the illuminance distribution:

## 4. Design of freeform projection optics

To accomplish this binocular waveguide display system that uses diffraction gratings as couplers, the projection optics at each side need to be specially designed. There are two requirements for the sizes of the collimated light in different fields. First, in the sagittal plane, the length of the collimated light need to be sufficiently large to ensure that the exit light from either side can completely cover both exit pupils. Second, in the tangential plane, the length of the collimated light in different fields should equal the corresponding distance *D* between the incident positions of two successive TIRs. The distance *D* is given by:

*d*is the thickness of the waveguide, and $\theta \text{'}$ is the diffraction angle in the YZ plane. Figure 7 illustrates the situations that when the in-coming light is wider than

*D*, the extra part out of

*D*will be diffracted multiple times by the in-coupler and then radiated out of the waveguide; when the in-coming light is narrower than

*D*, there will be gaps between the exit light beams.

To satisfy the requirements mentioned above, the projection optics are designed based on the waveguide system we proposed. In addition, an equivalent exit pupil is introduced to substitute the two separated ones, as illustrated in Fig. 8. $EP{D}_{Y}$ is 12 mm and $EP{D}_{X}$ is 5 mm. The diameter of the equivalent exit pupil $EP{D}_{E}$ is about 16.8 mm. The green lines denote the boundaries of the exit light in the marginal fields (–16.5°H, $\pm $12°V) in the XY plane. $\phi \text{'}$ is the maximum incline angle in the horizontal direction and is approximately 10°. Figure 8 indicates that if the exit light is sufficiently large in size to cover the equivalent exit pupil, it can exactly cover the two separated ones. In the optical design, we employ a FFS prism as the projection optics and adopt the convention of tracing the system backward, namely from the eye position to the micro-display. The waveguide, which transmits the collimated light from the equivalent exit pupil to the FFS prism, folds the optical paths in the tangential plane, such that the sizes of collimated light in different fields are modulated to the corresponding *D* given by Eq. (9). Thus, the two requirements can be satisfied.

The layouts of the waveguide system in the sagittal and tangential planes are presented in Figs. 9(a) and 9(b), respectively. The system is set to be symmetric about the XY plane. The front surface of the planar waveguide is denoted as 1 and the surfaces of the prism are denoted as 2, 2’, 3, and 4, respectively. 2 and 2’ are the two sides of the same surface situated close to the waveguide. Light emitted from the micro-display is first refracted by surface 4 and is then reflected by surfaces 2’ and 3. Subsequently, it is transmitted through surface 2 and is finally inserted into the waveguide from surface 1. The incident angle on surface 2’ must be larger than the critical angle of TIR. The structure control method is illustrated in the right subfigure of Fig. 9(a). The top marginal ray, ${R}_{u}$, of the maximum field in the positive *Y* direction and the bottom marginal ray, ${R}_{b}$, of the maximum field in the negative *Y* direction are traced. ${P}_{u2}$, ${P}_{u3}$, ${P}_{u2\text{'}}$, and ${P}_{u4}$ label intersection points of the ray ${R}_{u}$ and surfaces 2, 3, 2’ and 4, respectively; ${P}_{b2}$, ${P}_{b3}$, ${P}_{b2\text{'}}$, and ${P}_{b4}$ label intersection points of the ray ${R}_{b}$ and surfaces 2, 3, 2’ and 4, respectively. ${\theta}_{u2\text{'}}$ is the incident angle of the ray ${R}_{u}$ on surface 2’, which is the smallest one among all the incident angles. We choose the material PMMA with a refractive index *n* of 1.495 as the material of the prism because of its low density. Based on the physical structure as well as the propagating paths of the rays, the relationships of the parameters above are expressed as

Furthermore, a 0.61-in (1.5494-cm) micro-display with 1024 × 768 pixels is used as the image source. The representations of all FFSs are XY polynomials. With the constraints above, the parameters of the surfaces are optimized by the software CODE V. After optimization, the modulation transfer function (MTF) and distortion grid are presented in Figs. 10(a) and 10(b). As we can see, all MTF values are above 0.3 at 33 lp/mm, which is sufficient for the visual system. In Fig. 10(b), the distortion in the micro-display path is up to 8.1% at the right top/bottom corners and can be cancelled out by electronically pre-compensating the input images.

The overall layout of the binocular waveguide display system we proposed in the tangential plane is illustrated in Fig. 11. The rays in the five fields across the exit pupils are labelled ${F}_{1}$, ${F}_{2}$, ${F}_{3}$, ${F}_{4}$, and ${F}_{5}$, and they are from the pixels ${a}_{l}$, ${b}_{l}$, ${c}_{l}$, ${d}_{l}$ and ${e}_{l}$ on the left micro-display, respectively, and ${a}_{r}$, ${b}_{r}$, ${c}_{r}$, ${d}_{r}$ and ${e}_{r}$ on the right micro-display, respectively. It should be noted that the marked pixels on the two micro-displays are in reversed sequences. Therefore, the images displayed on the two screens should be upside down in the tangential plane to project the same image received by the eyes.

## 5. Discussions and conclusions

Regarding the fabrication, there might be a slight variation around the design values. A variation of $\pm $5% in the wavelength of the in-coming light and the period, the groove depth, and the film thickness of the gratings has been taken into consideration. The groove depth of the in-coupling grating and the out-coupling grating in the design is about 70.0 nm and 25.0 nm, respectively. The effects of the slight variations on the efficiency distributions and the illuminance uniformity of the combined waveguide are presented in Table 2. When the in-coming light has a dispersion of $\pm $5%, for all wavelengths across this spectrum, the efficiency uniformities of the useful orders are above 0.76 and illuminance uniformities of different fields are above 0.76. When the parameters of gratings change within $\pm $5%, the efficiency uniformities of the useful orders are above 0.78 and illuminance uniformities of different fields are above 0.75.

In realization, the thickness of the waveguide and the placement of the projection optics need to be slightly adjusted based on the variations of the period *T* of the gratings and the wavelength $\lambda $. From the grating equations, we can know that a shorter period *T* of grating couplers or a longer wavelength $\lambda $ leads to a larger diffraction angle. Correspondingly, the distance *D* between the incident positions of two successive TIRs will become longer. Thus, more collimated light from the projection optics will be coupled into waveguide and the MTF values of the system will decrease in such situations. We can choose a thinner waveguide to adjust the distance *D*, and to thus improve the quality of projected images. And when the period *T* of grating in-coupler slightly differs from that of the grating out-coupler, the angles of in-coming light and exit light will be different. To obtain a symmetrical FOV, we need to rotate the right projection optics clockwise and the left projection optics counter clockwise in the tangential plane when the period *T* of grating in-coupler is longer, and vice versa.

In conclusion, a monochromatic binocular waveguide display system with diffraction gratings and freeform optics is presented in this paper. Two surface-relief diffraction grating couplers are designed and the diffraction efficiencies of the couplers can remain stable when the incident angle varies in a broad angular range. This feature benefits the system with a wide diagonal FOV of 40° in diagonal. According to the symmetry structure of the grating out-coupler, two separated waveguides in the monocular subsystems are combined to improve the uniformity of illuminance over the expanded eye boxes. A FFS prism matching this binocular waveguide system is designed as the projection optics. The diameters of the two exit pupils are 12 mm in the expanding direction at an eye relief of 19 mm. This uniform-illumination design with a wide FOV is promising for applications in various binocular AR devices.

## Funding

National Natural Science Foundation of China (NSFC) (61475113).

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