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Abstract
A two-dimensional geometrical waveguide enables ultra-thin augmented reality (AR) near-eye display (NED) with wide field of view (FOV) and large exit-pupil diameter (EPD). A conventional design method can efficiently design waveguides that meet the requirements, but is unable to fully utilize the potential display performance of the waveguide. A forward-ray-tracing waveguide design method with maximum FOV analysis is proposed, enabling two-dimensional geometrical waveguides to achieve their maximum FOV while maintaining minimum dimensions. Finally, the designed stray-light-suppressed waveguide NED has a thickness of 1.7 mm, a FOV of 50.00°H × 29.92°V, and an eye-box of 12 mm × 12 mm at an eye-relief of 18 mm.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Augmented reality (AR) technology is capable of displaying virtual scenes generated by computer without blocking realistic scenes in real time, enabling intuitive and efficient information display [1]. Therefore, since it was first proposed in 1968 [2], AR technology has developed rapidly and has been widely used recently in various fields such as medical, education and entertainment [3]. AR near-eye display (NED) is the key hardware for AR technology to achieve virtual-reality fusion. Many companies have released their products in recent years, such as Microsoft's HoloLens [4], Lumus [5], MagicLeap [6], Waveoptics [7], and LingxiAR [8]. Related studies have proposed many solutions for NEDs, such as prisms [9], free-form surfaces [10], Birdbath [11], retinal projection [12], waveguides [13,14], metalens [15], and so on. However, there is no fully satisfactory solution that can achieve a large exit pupil diameter (EPD), small thickness, and large field of view (FOV) at the same time.
Traditional optical display solutions represented by free-form surfaces and prisms can achieve large field of view angles, but are typically more than 10 mm thick [16,17]. Retinal projection displays [12] can achieve large FOV without accommodation-vergence conflict, but require the user to precisely align the pupil to the convergence point of the Maxwell viewing method. The AR NEDs based on metalens can achieve large FOV displays [15], however, it is hard to solve the chromatic aberration of metasurface displays and the fabrication of metalenses is limited to the millimeter level.
Optical waveguides can achieve both large FOV and compact size. According to the principles of input and output couplers, optical waveguides can be divided into geometric waveguides [18,19] and diffractive waveguides [20,21]. Due to the constraints of diffraction principles, diffraction waveguides have obvious color distortion and serious light leakage. Geometrical waveguides offer a viable solution to the problem of chromatic dispersion while maintaining excellent imaging quality. Consequently, they have emerged as one of the most promising alternatives for consumer-grade AR glasses.
Most of the research on geometrical waveguides has focused on one-dimensional (1D) geometrical waveguides [13,14]. One-dimensional waveguides expand the exit pupil only in one direction [18,19]. To obtain an acceptable eye-box, the projection system should have a large exit pupil in the direction without expansion. As a consequence, the volume of the projection system should be larger, the width of the waveguide should be larger, and the image quality is greatly degraded compared with small exit pupil systems. Compared to two-dimensional (2D) waveguides, the design and manufacturing of one-dimensional geometrical waveguides are relatively simple. However, they have smaller fields of view, eye-relief, and exit pupil diameter.
In 2005, Yaakov Amitai et al. proposed the concept of the two-dimensional pupil-expanding geometrical waveguide for the first time [22]. In 2018, Gu Luo et al. proposed a design method for a two-dimensional stray-light-free geometrical waveguide head-mounted display [23]. The designed head-mounted display used two one-dimensional waveguides to achieve pupil expansion in two directions and optimized the stray light and illumination uniformity. However, the design with separated vertical and horizontal waveguides requires a long exit pupil distance of the projection optics, which limits the field of view. Moreover, the design has a complex structure and a large thickness. In 2020, Lumus Ltd. proposed a near-eye display system with a single-layer two-dimensional pupil expansion geometrical waveguide, which integrated the horizontal and vertical waveguides into the a single-layer waveguide [24]. In 2022, Cheng et al. designed a wide-angle, single-layer, two-dimensional pupil expansion geometrical waveguide, and completed the manufacturing and testing of the prototype, achieving a field of view of $45.2^\circ H \times 34.6^\circ V$ and an EPD of $12.0\; mm \times 10.0\; mm$ at an eye relief of $18.0\; mm$ [25]. The above research indicates that the single-layer two-dimensional waveguide NED device has the advantages of a large field of view, a compact shape, and a large EPD, and can achieve high-quality image display.
However, conventional design methods for two-dimensional geometrical waveguides have difficulty fully exploiting the display performance of the waveguide. The conventional methods design the waveguide inversely, which infers the shape and size that the waveguide should have from the required FOV, EPD and eye-relief. Conventional methods can quickly design the waveguide to meet the parameter requirements, but usually, the same structure of the waveguide can achieve better performance by adding an offset angle for the in-coupling prism and modifying the slanted angles of the waveguide. Once the design is complete, there is still a compromise to be made between wearing comfort and optical performance, meaning that it is difficult to completely avoid the situation that the designed waveguide size is not suitable for users to wear and use. How to improve FOV and reduce volume at the same time has become one of the most important issues for geometrical waveguides.
In our work, we proposed a design method for a single-layer two-dimensional pupil expansion geometrical waveguide with low stray light, large FOV, and large EPD based on the forward-ray-tracing and maximum FOV. The vertical region was designed first with the analysis of the constraints for the maximum vertical FOV. Tracing from the projection optics to the waveguide, the propagation path of a randomly given field in FOV was accurately calculated and simulated, and the process of exit pupil matching for maximum FOV optimization was demonstrated. By combining the limitations of the vertical FOV angle and the exit pupil matching optimization, the maximum FOV that can be achieved by the given two-dimensional geometrical waveguide can be found. The projection optics for the 2D geometrical waveguide near-eye display system were designed and integrated. Finally, a 2D waveguide with an $12.0\; mm\; \times \; 12.0\; mm$ eye-box, an $18.0\; mm$ eye-relief, a $50.00^\circ H\; \times \; 29.92^\circ V$ FOV, and a thickness of $1.7\; mm$ was designed and validated in the non-sequential ray tracing software LightTools. The results prove that for a given size of waveguide, the proposed waveguide design method with forward-ray-tracing and maximum FOV analysis enables a two-dimensional geometrical waveguide to achieve its maximum FOV while keeping the size as small as possible. The propagation path of ray bundle in the waveguide has been accurately described, providing a key theoretical basis for the improvement of waveguides in future.
2. Geometry configuration of two-dimensional geometrical waveguide
The 2D geometrical waveguide NED consists of a projection optics and the waveguide. The projection optics is composed of a micro-display and collimating optics, while the waveguide is composed of horizontal pupil expansion region (horizontal region), vertical pupil expansion region (vertical region) and an in-coupling prism. The in-coupling prism couples the collimated light from the projection optics into the waveguide. As a light propagates inside the waveguide, it encounters partially reflective mirror arrays (PRMAs), which expands light into multiple beams and exits in the same direction, thereby achieving pupil expansion. Figure 1 shows a schematic diagram of the 2D geometrical waveguide.
As shown in Fig. 2(b), the light emitted from the centermost pixel of micro-display is emitted perpendicular to the projected optics and is called the central field (indicated by the green line in Fig. 2), while the light emitted from the edge pixels of the micro-display is at the maximum angle to the central field and is called the marginal field (indicated by the blue and red line in Fig. 2). In the initial structure, as shown in Fig. 2(b), the in-coupling prism is parallel to the side of the waveguide, and thus the central field propagate along the X-axis and the marginal fields are symmetrical along the X-axis. After the pupil expansion in both directions, the exit pupil of the near-eye display is expanded from the smaller, unobservable exit pupil diameter to a larger, longer, exit pupil diameter ($EP{D_x} \times EP{D_y}$) suitable for user to observe, called the eye-box. Users can observe the displayed image within the eye-box of the near-eye display system.
The refractive index of waveguide is n. The length of the waveguide is L, and the width (along the Y direction) of the horizontal and vertical regions are ${w_h}$ and ${w_v}$, respectively. The thickness of the waveguide is d. The slanted angles of the horizontal and vertical PRMA are ${\theta _h}$ and ${\theta _v}$, respectively. The width of the in-coupling prism is ${w_p}$, which is determined by the EPD of the projection optics. The distance parameters of the partially reflective mirrors in the horizontal region are ${t_0},{t_1},{t_2}, \cdots ,{t_{i - 1}}$, where i is the total number of mirrors in the horizontal region. The distance between the partially reflective mirrors in the vertical region is H. A larger H ensures better transmission observation, which is usually determined by d and ${\theta _v}$, and the relationship is as follows:
3. Design of vertical region for maximum vertical FOV
The three critical performance parameters for NEDs are FOV, EPD, and eye-relief. The design of the projection optics determines FOV, while the design of the waveguide determines the size of eye-box (EPD) and eye-relief. Typically, the EPD and eye-relief are the first to be determined during the design process of waveguide. If the waveguide is designed with a larger eye-box, the user can still observe the displayed image clearly even when there is a slight displacement relative to the waveguide. Therefore, the size of the eye-box is usually designed to be 8-14 mm. The eye-relief of a NED is typically designed to be 16-20 mm, so that users can use the NED as if they were wearing glasses.
To fully realize the display performance of waveguide, after determining the EPD and eye-relief, analyzing and finding the maximum FOV becomes the most critical step in the waveguide design process. In Y-direction, the maximum FOV is limited by the eye-box conditions, total internal reflection (TIR) conditions, and stray light suppression conditions.
3.1 Eye-box constrain
The vertical FOV of the system is expressed as $FO{V_v} = 2{\mathrm{\Omega }_{v\; max}}$, where ${\mathrm{\Omega }_v}$ represents the angle of the vertical field in air, ${\mathrm{\Omega }_{v\; max}}$ is half of the vertical FOV in air, satisfying ${\mathrm{\Omega }_v} \in [{ - {\mathrm{\Omega }_{v\; max}},{\mathrm{\Omega }_{v\; max}}} ]$, and defining angle in the counterclockwise direction as positive. The size of the eye-box is denoted as $EP{D_x} \times EP{D_y}$, and the eye-relief is denoted as $ERF$. In the vertical direction (Y direction), its relationships can be expressed as:
3.2 TIR condition
As shown in Fig. 3, when the light at an angle of ${\mathrm{\Omega }_v}$ is coupled into the waveguide, according to the law of refraction, the refraction angle inside the waveguide is
where ${\omega _v} \in [{ - {\omega_{v\; max}},{\omega_{v\; max}}} ]$, defining the angle ${\mathrm{\omega }_v}$ is positive if the light is in the counterclockwise direction of the central field. When propagating inside the waveguide, the light must satisfy the total internal reflection condition. When reflecting on the top and bottom surfaces of waveguide, the angle of incidence of the field ${\mathrm{\Omega }_v}$(${\mathrm{\Omega }_v} \in [{ - {\mathrm{\Omega }_{v\; max}},{\mathrm{\Omega }_{v\; max}}} ]$) is $2{\theta _v} + {\omega _v}$. The TIR condition is satisfied, i.e., the angle of incidence is greater than the critical angle of TIR:When ${\omega _v}$ equals $- {\omega _{v\; max}}$, the incident angle reaches minimum value $2{\theta _v} - {\omega _{v\; max}}$, which should be larger than the critical angle of TIR. Therefore, according to the TIR condition, the second constraint of the vertical FOV can be deduced, as shown:
3.3 Stray light suppression condition
Wang et al. proposed in their work [19] that stray light from an 1D waveguide can be classified into three types: stray light due to the in-coupling structure, stray light due to abnormal reflections from the front surface and rear surface of the partially reflective mirror.
In our design, two of the three types of stray light can be generated in the vertically region of the 2D waveguide: stray light due to abnormal reflections from the front surface and rear surface of the partially reflective mirror. When the stray light propagates and meets the next partially reflective mirror, it is either reflected by the partially reflective mirror and coupled out, becoming stray light reducing the imaging quality; or it is reflected at a large angle of incidence and transformed into regular light again. In the stray light analysis, only the stray light that is successfully coupled out needs to be considered. As shown in Fig. 4, the angle of out-coupled stray light is inversely proportional to the half-FOV, and 2 types of stray light both can be described using a single equation:
where $\varepsilon $ represents the angle of incidence when stray light is refracted out of the waveguide, and defining the clockwise direction as positive.
Fig. 4. Stray light analysis of vertical FOV (a) stray light from different fields (b) exit pupil analysis of stray light.
According to (Eq. (7)) we can design the waveguide to direct stray light outside the eye-box, directly avoiding the stray light to be observed. When ${\omega _v}$ takes its maximum value ${\omega _{v\; max}}$, the out-coupling angle $\mathrm{{\rm E}}$ of stray light reaches the minimum value ${\mathrm{{\rm E}}_{min}}$, when the out-coupled stray light is closest to eye-box. In general, stray light appears only at the second partially reflective mirror. As shown in Fig. 4(b), when the stray light reaches the boundary of the eye-box, the angle between the stray light and the eye-box is called critical angle, denoted by $\delta $, and we have:
3.4 Maximum vertical FOV
Based on the above derivation, we can calculate the maximum vertical FOV under the eye-box constrain, TIR condition and stray light suppression condition. As shown in Fig. 5, the horizontal axis is the slanted angle of partially reflective mirror ${\theta _v}$ in the vertical region, and the vertical axis is the field ${\mathrm{\Omega }_v}$ of the incident light in the air. The lines in Fig. 5 represent the above three conditions and the restrictions of the field that ${\mathrm{\Omega }_v}$ must be greater than 0. The safe zone in Fig. 5 is the stray-light-free imaging area, and the uppermost point of the safe zone is the maximum vertical FOV. For example, when refractive index $n = 1.65$, the size of eye-box $EP{D_x} \times EP{D_y} = 12 \times 12mm$, the eye-relief $ERF = 18mm$, the thickness $h = 1.7mm$, the number of partially reflective mirrors $m = 7$, the bias of the eye-box $EY{E_{bias\; y}} = 2mm$, as shown in Fig. 5, it can be seen that when ${\theta _v} = 24.5^\circ $, it has ${\mathrm{\Omega }_{v\; max}} = 19.54$, that is $FO{V_{v\; max}} = 39.08^\circ $, that is, the maximum vertical FOV of this waveguide can reach $39.08^\circ $.
4. Forward-ray-tracing design with maximum FOV optimization
Analyzing the propagation path of rays plays a significant role in the design and optimization process of waveguide. It not only influences the dimensions of the waveguide, but also determines the maximum FOV of the waveguide. Since the in-coupled angles of the ray bundles are different, the propagation path of ray bundles varies for each different field of view. Tracing from the projection optics to the eye-box of the waveguide, forward-ray-tracing process enables us to numerically express the propagation path of every ray bundle. Thus, the exit pupil of the waveguide can be numerically determined.
4.1 Exit pupil of two-dimensional geometrical waveguides
The exit pupil of rays is determined by the propagation of the ray bundles. Figure 6(a) shows the top view of the propagation path of light from four marginal fields inside the waveguide. Figure 6(b) shows a single field (marginal field 3) propagating inside the waveguide, being reflected out of the waveguide, and finally reaching the exit pupil plane.
Fig. 6. Schematic (a) propagation path of four marginal fields in waveguide, (b) propagation and exit pupil of marginal field 3, (c) four in-coupled marginal fields and (d) the exit pupil matching of four marginal fields.
The area covered by the light in the exit pupil plane is called the exit pupil area of this field, and a human eye can observe this field when it is in the exit pupil area. In the exit pupil plane of the waveguide, each field covers a different area with different location and size, and the overlap of all exit pupil areas is the effective imaging area, in which user can observe the complete displayed image.
By adjusting the exit pupil areas of all fields, exit pupil matching finally makes the effective imaging area meet the requirements of the eye-box, as shown in Fig. 6(d).The process of pupil matching is an important step in the design of a two-dimensional geometrical waveguide, which requires designing a reasonable waveguide structure according to the propagation path of ray bundles.
4.2 Forward-ray-tracing
In order to numerically express the propagation of the ray bundles, the forward-ray-tracing is calculated. The horizontal and vertical fields in air of the in-coupled rays are denoted by $({{\mathrm{\Omega }_h},{\mathrm{\Omega }_v}} )$, as shown in Fig. 6(c). Consider the propagation and exit pupil of 4 marginal fields $({ \pm {\mathrm{\Omega }_{h\; max}}, \pm {\mathrm{\Omega }_{v\; max}}} )$ in the exit pupil plane: marginal fields 1, 2, 3 and 4 are represented by red, orange, green, and blue lines, respectively. When the exit pupil areas of four marginal fields cover the entire eye-box, the exit pupil area of all fields also cover the eye-box [24], at which time the user can observe the complete display image in eye-box. The slanted angle of the in-coupling prism is usually designed as $2{\theta _v}$ to make the out-coupled central field perpendicular to the waveguide surface. Figure 6(c) shows the schematic diagram of the incidence of the four marginal fields on the in-coupling prism, where the white line indicates the central field, the white dashed lines indicate the auxiliary lines perpendicular to the surface of the in-coupling prism, and the red lines indicate the incident and refracted marginal field 1.
Figure 7(a) represents the schematic diagram when any ray field $({{\mathrm{\Omega }_h},{\mathrm{\Omega }_v}} )$ is incident on the in-coupling prism, and the angle of incidence ${\mathrm{\Phi }_{in}}$ of the ray can be expressed as:
Fig. 7. Schematic (a) in-coupling process of a random field and (b) out-coupling process of a random field.
After refraction, the angle between the light inside the waveguide and the surface of the in-coupling prism
The field angle ${\mathrm{\varphi }_v}$ and ${\mathrm{\varphi }_h}$ of the refracted rays can be calculated by:
After entering the waveguide, the propagation of light can be described by 3 angles: the angle between the projection of light in XOY plane and the X-axis ${\mathrm{\alpha }_h}$ in the horizontal region (shown in Fig. 6(a)), the angle between the projection of light in XOY plane and the Y-axis ${\mathrm{\alpha }_v}$ in the vertical region (shown in Fig. 6(a)) and the angle between the light and XOY plane $\beta $ (shown in Fig. 7(a)). For a waveguide without an in-coupling prism offset angle, the angles can be calculated as:
In the vertical exit pupil expansion region, the light is reflected by the vertical PRMA and refracted out of the waveguide, as shown in Fig. 7(b). From the highest point of light (point A) as the starting point, draw a line AO perpendicular to the bottom surface. And set the point O as the origin of the local coordinate system with coordinates $({0,0,0} )$. The coordinates of point A are $({0,0,d} )$. Line AB is perpendicular to the plane of vertical partially reflective mirror. The normal vector perpendicular to the partially reflective mirrors is denoted by $\vec{{\boldsymbol n}}$, line AC is denoted by the vector $\vec{{\boldsymbol l}}$, and the reflected light of AC is denoted by $\vec{{\boldsymbol r}}$.
And we have $\vec{{\boldsymbol r}}$ expressed as:
The angle between the reflected ray $\vec{{\boldsymbol r}}$ and the top surface of the waveguide is denoted by ${\varphi _{out}}$, and the directions of $\vec{{\boldsymbol r}}$ and the out-coupled light are the same and the angle is denoted by ${\alpha _{out}}$:
In general, the waveguide is designed based on the principle that the out-coupled central field is perpendicular to the surface of the waveguide, and the focal power of waveguide is zero. Thus, the field out-coupled from the waveguide is the same as when it is incident. That is, the following relationship exists:
According to (Eq. (10))- Eq. (16), the complete propagation process of a random field $({{\mathrm{\Omega }_h},{\mathrm{\Omega }_v}} )$ in the waveguide can be calculated. Therefore, numerical simulations of forward-ray-tracing in a 2D geometrical waveguide can be performed to obtain the propagation path and exit pupil area for all fields.
4.3 Exit pupil matching for maximum FOV optimization
In the design of waveguide, the three key performance parameters of the waveguide: FOV, EPD, and eye-relief constrain each other, and an increase in one parameter leads to decreases in the other parameters. Usually, the eye-relief and EPD of the waveguide are firstly determined to meet the design requirements. In the exit pupil matching process, the maximum FOV of the waveguide can be achieved by minimizing the additional effective imaging area outside the eye-box.
The numerical simulations of the exit pupil matching in a 2D waveguide can be carried out to optimize the design. In the 2D waveguide, the specific propagation paths of the marginal fields are shown in Fig. 8(a) and (b). The exit pupil areas of four marginal fields are calculated by numerical simulation and the results are shown as color-filled parallelograms in Fig. 8(c) and (d). In Fig. 8(c), it is shown that when the in-coupling prism is parallel to the side of the waveguide, the incident marginal fields are horizontally symmetrically distributed, and the effective imaging area of the exit pupil plane does not completely cover the eye-box. This is because the green and blue fields cannot propagate to the left side of the waveguide, which in turn results in a smaller coverage area in the exit plane. Therefore, in Fig. 8(d), the in-coupling prism is given an offset angle ${\theta _p}$, and the clockwise direction is noted as positive. In this situation, both green and blue fields reach the left side of the waveguide, and the exit pupil areas of them cover the eye-box. Thus, in this 2D waveguide, the exit pupil areas of all fields are matched together.
Fig. 8. (a) Top view of the exit pupil path of marginal fields, (b) perspective view, (c) numerical simulated exit pupil matching result (without offset angle of in-coupling prism) of four marginal fields and (d) with offset angle.
In order to ensure that the out-coupled central field is perpendicular to the waveguide, when introducing an offset angle to the in-coupling prism, the slanted angle of the partially reflective mirrors in the horizontal region should be:
In the exit pupil matching process, the matching result is related to many parameters: the size of the waveguide L, the width of the horizontal region ${w_h}$, the thickness of the waveguide d, the width of the in-coupling prism ${w_p}$, the position and angle of the in-coupling prism, the angle of the vertical PRMA ${\theta _v}$, and other parameters. As the FOV increases, the effective imaging area of the waveguide shrinks dramatically and eventually cannot completely cover the eye-box, as shown in Fig. 9. The parameters of exit pupil matching in the example shown in Fig. 9 are: refractive index $n = 1.65$, eye-relief $ERF = 18$, eye-box $EPD = 12 \times 12$, width of the in-coupling prism is ${w_p} = 4$, and the slanted angle of PRMA in the vertical region is ${\theta _v} = 25^\circ $. According to the above analysis shown in Fig. 5, when ${\theta _v} = 25^\circ $, the maximum vertical FOV without stray light is $30.22^\circ $. In Fig. 9, $Pris{m_{bias\; y}}$ indicates the distance bias of the in-coupling prism in the Y-axis direction, and the direction of Y-axis is taken as positive, and the in-coupling prism is at the center of the horizontal region when $Pris{m_{bias\; y}} = 0$.
Fig. 9. Numerically simulated exit pupil matching results of a same waveguide to find its maximum FOV.
According to Fig. 9, the increase of the horizontal FOV mainly leads to the decrease of length in X direction of effective imaging area, and the increase of the vertical FOV mainly leads to the decrease of width in Y direction of effective imaging area. Increasing the length of the waveguide L and offset angle of prism ${\theta _p}$ can effectively expand the effective imaging area, while increasing the width of the vertical region ${w_p}$ can significantly increase the width in Y direction of the effective imaging area. Therefore, the maximum FOV analysis of the waveguide is done by exit pupil matching, which is a process of iterative optimization with multiple parameter changes. The design method of maximum FOV analysis can find the maximum FOV of any 2D waveguide and improve the performance of NED without changing the size.
5. Design of projection optics and system integration
In a waveguide NED, a projection optics is required to magnify the image of the micro-display and project it into the in-coupling prism of the waveguide. Since the focal power of waveguide is zero, the FOV of the waveguide NED system equals to the FOV of the projection optics. Therefore, in order to maximize the performance of waveguide, the FOV of the projection optics should be set to the maximum FOV of waveguide.
The slanted angle of vertical PRMA was selected as ${\theta _v} = 25^\circ $, corresponding to the stray-light-free maximum vertical FOV of $30.22^\circ $ shown in Fig. 5. The display image ratio was set to $16:9$, so the FOV of the system should be $FOV = 50^\circ \; H \times 30\; ^\circ \; V$. After finishing exit pupil matching, the configuration parameters of the waveguide are listed in Table 1.
Table 1. Geometrical configuration of designed waveguide
In this work, a $0.39^{\prime \prime}$ micro-display was selected as the image source. The active size of the micro-display was $8.75mm \times 4.97mm$ and the resolution was $1920 \times 1080$. Based on the configuration of waveguide in Tab.1, we designed the projection optics containing spherical surfaces, as shown in Fig. 10(a). The length ${L_p}$ of the projection optics was $15.3mm$. The FOV and the exit pupil diameter of the projection optics were $56.5^\circ $($50.00^\circ H \times 29.92^\circ V$) and $4mm$ with an exit pupil distance of $0.5mm$. The effective focal lengths (EFL) and F/# were $9.33mm$ and $2.31$. Figure 10(b) shows the spot diagram of the designed projection optics, Fig. 10(c) shows the distortion grid, and the maximum value is less than $1\%$, and Fig. 10(d) shows the modulation transfer function (MTF) curves which are all above $0.5$ at $56lp/mm$ for all fields.
Fig. 10. (a) Optical layout of projection optics, (b) spot diagram, (c) distortion grid, (d) MTF diagram.
The 2D waveguide, the projection optics and the micro-display were integrated into a compact glass-like NED system, demonstrating that the designed 2D waveguide NED system was free from the disadvantages of the conventional NED system which was larger and bulkier. The integrated NED system was simulated in LightTools simulation software to verify the proposed design method, as shown in Fig. 11. Figure 11(b) shows the integrated two-dimensional waveguide NED system, where the light emits from the rectangular micro-display, is collimated by the projection light path and then incident into the in-coupling prism. Then the light undergoes pupil expansion inside the waveguide before exiting the waveguide. Finally, the light is expanded into a large area of parallel light propagating toward eye-box.
Fig. 11. (a) Exit pupil matching result of the designed waveguide, (b) system integration for verification simulation.
We performed exit pupil matching to find the maximum FOV that the waveguide can achieve. Figure 11(a) shows the results of the exit pupil matching: the effective imaging area of four marginal fields in the exit pupil plane completely covers the eye-box, and the maximum FOV of the waveguide is $50.00^\circ H \times 30.00^\circ V$.
To verify the proposed exit pupil matching method of two-dimensional geometrical waveguide for all fields, a receiver is placed on the exit pupil plane. Four light sources were placed at the four corners of the micro-display to simulate four marginal fields. The exit pupil areas of four marginal fields in the exit plane is shown in Fig. 12, and the results are consistent with the image obtained from the exit pupil matching calculation (Fig. 11(a)). In this case, the boundaries of the exit pupil areas of three marginal fields just cover the eye-box, so the FOV of the waveguide reaches maximum value, i.e., the performance of the waveguide is brought into full play.
An ideal lens with an aperture of $4mm$ was placed at an eye-relief distance of $18mm$ from the waveguide surface, and a receiver was placed at the focal plane of the ideal lens. The focal length of the ideal lens is $16mm$, similar to that of the human eye, in order to simulate the observation of a human eye and examine the stray light of the waveguide. As shown in Fig. 13(a), nine-point light sources were set on the surface of the micro-display, and the light from each source was imaged on the receiver after passing through the ideal lens. The results are shown in Fig. 13(b), where each light source is imaged as a dot in the receiver plane, proving that the pixels on the micro-display in the NED system can be clearly observed by a human eye, i.e., the NED system can realize the projection display of virtual images.
Fig. 13. (a) Simulation of eye observation of the designed NED, (b) the propagation of crosstalk stray light and the simulation result.
There is no stray light from the vertical region because the vertical FOV does not exceed the maximum vertical FOV analyzed. However, there is still some crosstalk stray light in the waveguide, as shown in Fig. 12(a), (b) and Fig. 13(b). The cause is that the offset angle of in-coupling prism is introduced in the optimization process, so that the light from one side is incident at a large angle to the vertical region. Then the light is reflected by the vertical PRMA and returns to the horizontal region, but is soon reflected again. This unexpected process changes the direction of the light and causes stray light. The crosstalk stray light can be partially suppressed by adding a tri-prism in the horizontal region, as shown in Fig. 11(b). Then the crosstalk stray light only exists near the marginal field, but it is difficult to completely eliminate. Avoiding a large offset angle of the in-coupling prism is also a way to suppress crosstalk stray light, but this will limit the FOV of the waveguide.
6. Conclusion
In this work, a design method with forward-ray-tracing and maximum FOV analysis of the single-layer two-dimensional pupil expansion geometrical waveguide was proposed. The geometry configuration of the two-dimensional geometrical waveguide, design principles of vertical region, forward-ray-tracing design method and exit pupil matching optimization for maximum FOV were presented. A single-layer 2D waveguide and a compact projection optics with the maximum FOV of the waveguide were designed and integrated. Finally, the designed 2D geometrical waveguide had a thickness of $1.7mm$, a FOV of $50.00^\circ H \times 29.92^\circ V$, and an eye-box of $12mm \times 12mm$ at an eye-relief of $18mm$. All of the stray light from vertical pupil expansion region and most of the crosstalk stray light were suppressed. The simulation results demonstrated that the designed waveguide with determined size have reached its maximum FOV and showed that the illumination uniformity of the different fields is acceptable. Most of the stray light was separated from the regular light, and only the crosstalk stray light near the marginal field can be observed by the simulated human eye.
For a given size of waveguide, the proposed 2D waveguide forward-ray-tracing design method with maximum FOV analysis can fully utilize the display performance of the waveguide, enables the two-dimensional geometrical waveguide to achieve the maximum FOV while keeping the size as small as possible. Usually, the shape size and exit pupil size of a consumer AR waveguide should meet the viewing habits of human and also have excellent optical performance. The propagation path has been accurately described, providing a key theoretical basis for the improvement of waveguides in future. The design method proposed in this work provides a possible solution for consumer AR devices to meet both ergonomic requirements and optical performance requirements.
Funding
National Key Research and Development Program of China (2021YFB3701504, 2021YFC2202403); Strategic Priority Research Program of the Chinese Academy of Sciences (XDA25020317); National Natural Science Foundation of China (62175259).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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