Uniformity improvement of two-dimensional surface relief grating waveguide display using particle swarm optimization

Dongwei Ni; Dewen Cheng; Dewen Cheng; Yue Liu; Ximeng Wang; Cheng Yao; Tong Yang; Cheng Chi; Yongtian Wang

doi:10.1364/OE.462384

1. Introduction

Augmented reality head-mounted displays (AR-HMDs) are developing rapidly as a type of next-generation display and have a wide range of applications in the military, education, entertainment, and navigation [1]. Several methods can be applied to realize AR-HMDs, including freeform surface prism systems [2,3], projection systems [4], off-axis reflective-refractive systems [5,6], and optical waveguides [7–9]. Optical waveguides are the most promising owing to their glass form, which improves the portability and wearability of AR devices. In general, an optical waveguide includes an in-coupling structure and an out-coupling structure. According to the working principle of the coupling structure, waveguide technology can be divided into geometric waveguides [10–13] and diffractive waveguides [14–18]. Refractive or reflective optics are commonly used in geometric waveguides as the in-coupler and out-coupler. However, it suffers from the problems such as a small exit pupil, low transmittance, and low yield rate. The maximum field of view (FOV) developed by Google was less than 20°, and the exit pupil range was also small [19]. A reflective mirror array was also used as an out-coupler by Lumus [12,20], and a multi-layer glue coating that requires precise bonding was used, resulting in low yields.

The in-coupling and out-coupling optical elements of the diffractive waveguide adopt a periodic grating structure, which can be further divided into surface relief grating (SRG) waveguide and volume holographic grating (VHG) waveguide according to the difference in the periodic structure. Microsoft [9] and Magic Leap [21] developed AR-HMD products with diffractive waveguide using SRG, BAE developed a Q-sight holographic waveguide based on VHG [22], and Sony developed a high-brightness transmissive AR-HMD using multi-layer VHGs [23]. However, AR-HMDs based on VHGs are disadvantaged by low diffraction efficiency, serious color crosstalk, and angular and wavelength selectivity dependence [24,25]. Owing to the angular bandwidth limitation of the VHG, it is difficult to ensure constant diffraction efficiency in different FOVs, which leads to poor illuminance uniformity of the waveguide display. Therefore, the optimal design scheme in this study utilizes an SRG diffractive waveguide. The designed gratings work at subwavelength and can achieve the required diffraction efficiency at the specified diffraction order, such as the literatures [26,27].

Exit pupil size and image illuminance uniformity are important factors that affect the optical performance of waveguide display systems. Several approaches to expand the exit pupil and improve illuminance uniformity have been proposed in previous studies. Liu and Pan et al. [28,29] expanded the one-dimensional (1D) exit pupil using diffraction grating technology. However, they only optimized the coupling grating without considering the overall illuminance uniformity of the waveguide as the premise of the optimization. The effects of different incident azimuth angles on the stability of the diffraction efficiency were also not considered. For the 2D-EPE diffractive waveguide, an important issue that affects the waveguide display is the illuminance uniformity, including the exit pupil illuminance uniformity and angular illuminance uniformity. A general approach [30,8] for gradually increasing the diffraction efficiency of the out-coupling grating has been proposed. This method is based on a simple proportional distribution of the diffraction efficiency to compensate for the illuminance uniformity at the exit pupil position, while the angular illuminance uniformity is not considered. The diffraction efficiency of the grating changes significantly with an increase in the FOV, which makes it more difficult to realize image illuminance uniformity. Nakamura et al. [31] presented illuminance uniformity by using discrete depth-varying holographic gratings. However, this scheme cannot realize the uniformity of the exit pupil illuminance and the stability of the grating diffraction efficiency with different FOVs.

In this study, an optimal design method for a high uniformity SRG waveguide with exit pupil expansion (EPE) is proposed. Firstly, a redirection grating (DOE2) is embedded between the in-coupling grating (DOE1) and the out-coupling grating (DOE3) of the waveguide, and the 1D beam propagation is converted to two-dimensional (2D) beam propagation; thus, the exit pupil can be expanded to achieve a 2D-EPE. Secondly, to achieve illuminance uniformity of the waveguide display, the diffractive waveguide is divided into several independent diffractive sub-regions, and an illuminance uniformity model is established based on the energy propagation process. Combined with the non-sequential ray tracing and uniformity model, the diffraction efficiency of each grating sub-region is optimized to modulate the energy distribution in each sub-region. Through the gradual adjustment of each sub-region diffraction efficiency, the uniformity of the exit pupil illuminance and angular illuminance is realized.

The diffraction efficiency of the grating in different sub-regions is obtained by optimizing the illuminance uniformity of the diffraction waveguide, which is the ideal diffraction efficiency. The actual diffraction efficiency of the grating changes with the incident and azimuth angles of different FOV. It is difficult to guarantee that the diffraction efficiency of the grating remains stable with the change in different FOVs only by optimizing the grating parameters using simulation software. Finally, we propose a grating optimization strategy based on the particle swarm optimization (PSO) algorithm [32,33]. By establishing a fitness function, the actual diffraction efficiency of the different FOVs is constrained to the ideal efficiency. The PSO algorithm is used to integrate rigorous couple wave analysis (RCWA) [34–36]. The optimal grating structural parameters are obtained by PSO iterative optimization to improve the stability of the diffraction efficiency in different FOVs, which further ensures the uniformity of the exit pupil illuminance and angular illuminance of the waveguide display.

2. 2D-EPE waveguide system design

2.1 EPE principles overview

As shown in Fig. 1, a straight-line 2D-EPE diffractive waveguide structure is proposed. The light beam is coupled to the waveguide plate through DOE1 and then propagates forward in the total internal reflection (TIR) mode. When the light beam propagates to DOE2, it continuously contacts DOE2, and 1D beam propagation is transformed into 2D beam cluster propagation. Part of the light beam continues to propagate in the original direction in DOE2, and other light beams are diffracted and propagated in the direction of DOE3. The beam expands in the vertical direction (the positive direction of the X-axis) before reaching the DOE3, and after the diffraction of the DOE3, the horizontal direction (the positive direction of the Y-axis) is also expanded and a 2D beam band is formed.

Fig. 1. Schematic of ray propagation in 2D-EPE diffractive waveguide.

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After the light beam propagation is repeatedly diffracted by DOE2 and DOE3, a part of the light beam is diffracted by DOE3 and emits from the waveguide. Therefore, the energy of the beam gradually decreases along the direction of the EPE, and the uniformity of the image illuminance is destroyed. Therefore, to meet the requirements of imaging uniformity, the diffraction efficiencies of DOE2 and DOE3 should be gradually increased along the propagation direction of the beam, and the diffraction efficiencies should remain constant when the incident FOV of the grating changes.

DOE1 modulates the incident light in the waveguide and controls the propagation energy. In this design, the energy of the diffracted light from DOE1 should concentrate mainly on its positive first reflective order (R₁). The light beam in R₁ is confined in the waveguide and propagated to DOE2 by TIR, and there are diffracted light beams of negative first reflective order (R_-1), zeroth reflective order (R₀), and R₁ in DOE2. There are mainly R_-1 and R₀ diffracted light beams in DOE3.

2.2 Initial parameters and profile of diffraction grating couplers

Light propagates through a certain diffraction order in the waveguide, and additional grating diffraction orders should be avoided. Therefore, the grating period needs to be optimized to ensure that light of the same incident FOV can propagate in the waveguide along a single-order direction. The diffraction efficiency of the symmetric order is uniformly distributed owing to the structural symmetry of the binary rectangular grating. Most of the light is diffracted out of the waveguide by the zero-order, resulting in lower diffraction efficiency of other orders. Thus, it is difficult to maximize the diffraction efficiency of DOE1. The slanted surface relief grating (SSRG) breaks the symmetry of the grating and can achieve high diffraction efficiency at a specific diffraction order. Based on this principle, all the gratings in this study are designed with an SSRG to achieve the efficiency of the specified diffraction order.

The diffraction efficiency of the grating is affected by the angle of incidence and its azimuth angle to the grating. As shown in Fig. 2, conical diffraction is considered when the light hits the coupling grating, where α is the slant angle, h is the groove depth, c is the groove width, d is the grating period, and the fill factor is f = (d - c)/d. The y-axis is the direction of the grating line, x-axis is the direction perpendicular to the grating line, and z-axis is the direction perpendicular to the grating surface. The incident angle θ is the angle between the incident wave vector-k and the z-axis, the incident azimuth angle φ is the angle between the projection vector of the incident wave vector on the xy-plane and the x-axis, and θ_m is the diffraction angle of the grating. Areas 1, 2, 3, and 4 represent the incident/reflection area, the grating modulation area, the base layer, and the transmission area, respectively. The refractive index of areas 1 and 4 are n₀= 1, and the refractive index of areas 2 and 3 are n.

Fig. 2. Diagram of SSRG structure and conical diffraction.

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The designed 2D-EPE diffractive waveguide uses reflective SSRGs, and in the case of conical diffraction, the vector-k of the incident light entering the waveguide can be described as [28,29]:

(1)$${{\boldsymbol k}_{\boldsymbol i}} = \frac{{2\pi }}{\lambda }{n_0}(\sin {\theta _0}\cos {\varphi _0},\sin {\theta _0}\sin {\varphi _0},\cos {\theta _0})$$

where λ is the wavelength of the incident light, and θ₀ and φ₀ are the incident and azimuth angles of the grating, respectively.

The vector-k of the m-th diffraction order in the waveguide is

(2)$${{\boldsymbol k}_{\boldsymbol m}} = \frac{{2\pi }}{\lambda }n(\sin \theta _\textrm{m}^\mathrm{^{\prime}}\cos \varphi _\textrm{m}^\mathrm{^{\prime}},\sin \theta _\textrm{m}^\mathrm{^{\prime}}\sin \varphi _\textrm{m}^\mathrm{^{\prime}},\cos \theta _\textrm{m}^\mathrm{^{\prime}})$$

where θ'_m and φ'_m are the diffraction and azimuth angles of the grating, respectively.

Grating diffraction equation under conical geometry:

(3)$$n\sin \theta _\textrm{m}^\mathrm{^{\prime}}\sin \varphi _\textrm{m}^\mathrm{^{\prime}} = {n_0}\sin {\theta _0}\sin {\varphi _0} = \gamma$$

(4)$$n\sin \theta _\textrm{m}^\mathrm{^{\prime}}\cos \varphi _\textrm{m}^\mathrm{^{\prime}} = {n_0}\sin {\theta _0}\cos {\varphi _0} + m\frac{\lambda }{d} = {\alpha _0} + m\frac{\lambda }{d}$$

To meet the requirements of the grating to generate first-order diffraction propagation and TIR in the waveguide, the period d should have the following constraints:

(5)$$d < \min \left\{ {\frac{\lambda }{{\sqrt {1 - {\gamma^2}} - {\alpha_0}}}, \frac{\lambda }{{\sqrt {1 - {\gamma^2}} \textrm{ + }{\alpha_0}}}} \right\}$$

The conical diffraction and propagation direction of light in the waveguide are shown in Fig. 3. DOE2 deflects the light propagation to achieve pupil expansion. To ensure that the FOV emitted from the waveguide is equal to the incident FOV, the periods of DOE1 and DOE3 are the same. In our design, the wavelength of the incident light is 532 nm, the FOV is set to 24° × 18°, the exit pupil size is 16 mm × 14 mm at an eye relief (ERF) of 20 mm, the waveguide substrate thickness is 1.2 mm, and its material is H-ZLAF3. The periods of DOE1 and DOE3 are 420 nm, and the period of DOE2 is 330 nm.

Fig. 3. (a) Division of diffractive waveguide grating regions, (b) light propagation in the edge positive FOV, (c) light propagation in the edge negative FOV, (d) light propagation in the middle part of FOV.

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2.3 Principle of the 2D-EPE diffraction

Because the diffracted light of DOE1 is continuously diffracted in DOE2 and DOE3, the light energy gradually weakens along the light propagation direction, which leads to an uneven distribution of illuminance in the exit pupil. To realize uniform distribution of the illuminance, it is necessary to divide DOE2 and DOE3 to achieve the gradual distribution of the grating diffraction efficiency, as shown in Fig. 3(a).

DOE2 is divided into two parts A and B, where each part is divided into four sub-regions, and DOE3 is divided into four regions. The light propagates in each sub-region as shown in Figs. 3(b–d). Parts A and B of DOE2 can accommodate the FOV in the same or different directions. The light propagation in the edge positive FOV is shown in Fig. 3(b), the edge negative FOV propagation is shown in Fig. 3(c), and the middle part of the FOV is shown in Fig. 3(d). DOE2 has grating vectors in different directions. As shown in the red box of Fig. 3(b), part B of DOE2 is composed of two grating vectors with opposite directions. The grating vector of sub-region 2 is k_B2. The grating vectors of sub-regions 3, 4 and 5 are k_B3, k_B4 and k_B5 with the same direction, respectively. The grating vector direction of sub-region 2 is opposite to that of sub-region 3, 4 and 5. Also in part A, as shown in the red box of Fig. 3(c), the grating vectors k_A7, k_A8, k_A9 has the same direction, and the direction of the grating vector k_A6 in sub-region 6 is opposite to that of the grating vectors k_A7, k_A8, k_A9 in sub-regions 7, 8, and 9.

DOE1 diffracted light propagates to sub-regions 2 and 6 of DOE2, where it is diffracted into R₀ and R_-1, respectively. The light in R₀ propagates to DOE3, the R_-1 diffracted light of sub-region 2 propagates to sub-region 3 of B, and the R_-1 diffracted light of sub-region 6 propagates to sub-region 7 of A. The R₁ diffracted light of the other sub-regions except sub-regions 2 and 6 propagates to DOE3. DOE3 is divided into four regions, which receive R₀ and R₁ light from DOE2, and its diffracted R_-1 light exits from the waveguide and enters the human eye.

In the waveguide, the light of different FOV propagates to the grating region with a specific incident angle and azimuth angle, and light with different incident angles entering to the same grating region has different diffraction efficiencies. Therefore, the high-precision calculation of the incident angle and azimuth angle of the entering light is a vital prerequisite for optimizing the parameters of the grating structure. As shown in Figs. 3(b−d), in the spherical coordinate system, the incident angle of light at DOE1 inside the waveguide is (θ₀, φ₀), and the angle of the diffracted light is (θ₁, φ₁). The period of DOE1 is d₁, and ρ is the rotation angle of the grating vector direction of DOE2 relative to that of DOE1. The diffraction angle and period of DOE2 are (θ₂, φ₂) and d₂, respectively. The diffraction angle of DOE3 is (θ₃, φ₃) and the period is d₃. The specific calculation process of the propagation inside the waveguide is expressed as:

(6)$$\sin {\theta _1}\sin {\varphi _1} = \sin {\theta _0}\sin {\varphi _0}$$

(7)$$n\sin {\theta _1}\cos {\varphi _1} = n\sin {\theta _0}\cos {\varphi _0} + \frac{\lambda }{{{d_\textrm{1}}}}$$

The DOE2 diffracts light to m-order, which satisfies:

(8)$$\sin {\theta _2}\sin {\varphi _2} = \sin {\theta _1}sin({\varphi _1} + \rho )$$

(9)$$n\sin {\theta _2}\cos {\varphi _2} = n\sin {\theta _1}\cos ({\varphi _1} + \rho )\textrm{ + m}\frac{\lambda }{{{d_\textrm{2}}}}$$

When light reaches the grating in DOE3, the diffraction angle of DOE3 should satisfy:

(10)$$\sin {\theta _3}\sin {\varphi _3} = \sin {\theta _2}sin({\varphi _2} - \rho )$$

(11)$$n\sin {\theta _3}\cos {\varphi _3} = n\sin {\theta _2}\cos ({\varphi _2} - \rho ) - \frac{\lambda }{{{d_\textrm{3}}}}$$

The incident angle θ and azimuth angle φ of each grating sub-region can be obtained using the above formula. The results are summarized in Table 1, which can be used to optimize the structural parameters of the grating to improve the stability of the diffraction efficiency in section 4.

Table 1. Incident angle and azimuth angle of each grating sub-region

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3. Optimization of illuminance distribution in 2D-EPE waveguide

The uniformity of image brightness in the exit pupil is an important factor for evaluating near-eye display performance. DOE1 couples the light emitted by the microdisplay into the waveguide to guarantee the highest possible diffraction efficiency. DOE2 and DOE3 are divided into different regions, and the diffraction efficiency of the different regions needs to be different such that uniform illumination across the exit pupil can be ensured. The diffractive waveguide structure can be modeled using the optical software LightTools. A non-sequential ray tracing simulation of the 2D-EPE-SSRG diffractive waveguide was performed, as shown in Fig. 4(a). The EPE in the x and y directions is shown in Figs. 4(b) and (c). The exit pupil sizes EPD_X and EPD_Y are 14 mm and 16 mm, respectively. The outgoing FOV in the x and y directions is |θ_V1| = |θ_V2 | = 9°and |θ_H1| = |θ_H2| = 12°.

Fig. 4. (a) Schematic illustration of the 2D-EPE-SSRG diffractive waveguide, (b) EPE ray path in X direction, (c) EPE ray path in Y direction.

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3.1 Optimization of illuminance uniformity

Exit pupil uniformity is defined as the uniformity of different exit pupil positions under the same FOV, and angular uniformity is defined as the uniformity of different FOV at the same exit pupil position. To realize the uniform illumination of an observed image, an illuminance uniformity evaluation model is established based on the propagation process of energy in the waveguide. The diffraction efficiency of the divided grating regions is optimized by non-sequential ray tracing method to realize a uniform illumination distribution at the entire 2D exit pupil.

The illuminance uniformity evaluation standards and optimization target constraints of the exit pupil uniformity and angular uniformity are expressed by Eqs. (12) and (13), respectively. In the optimization process, the diffraction efficiency of different sub-regions is used as a variable to balance the exit pupil uniformity and angular uniformity. Sampling optimization is performed for different exit pupil positions with the same FOV and the same exit pupil positions with different FOV by non-sequential ray tracing. P_max and P_min are the maximum and minimum illuminances of the same FOV at different exit pupil positions, Γ_P is the exit pupil uniformity. A_max and A_min are the maximum and minimum illuminances at the same exit pupil position in different FOV, Γ_A is the angular uniformity, ρ is the weight coefficient, and the optimization target value is set to 0.7.

(12)$$\left\{ {\begin{array}{c} {{\varGamma_\textrm{P}} = 1 - \frac{{{P_{\textrm{max}}} - {P_{\textrm{min}}}}}{{{P_{\textrm{max}}} + {P_{\textrm{min}}}}}}\\ {{E_{\textrm{pupil\_error}}} = \sum\limits_{i\textrm{ = 1},j = 1}^{i = {M_P},j = {N_P}} {\rho \ast {{({\varGamma_\textrm{P}}({x_i},{y_j}) - 0.\textrm{7})}^2}} } \end{array}} \right.$$

(13)$$\left\{ {\begin{array}{c} {{\varGamma_\textrm{A}} = 1 - \frac{{{A_{\textrm{max}}} - {A_{\textrm{min}}}}}{{{A_{\textrm{max}}} + {A_{\textrm{min}}}}}}\\ {{E_{\textrm{angular\_error}}} = \sum\limits_{i = 1,j = 1}^{i = {M_A},j = {N_A}} {\rho \ast {{({\varGamma_\textrm{A}}(i,j) - 0.\textrm{7})}^2}} } \end{array}} \right.$$

The sampling of the exit pupil position and FOV position is shown in Figs. 5(a) and (b), respectively. The exit pupil position is divided into M_p× N_P regions, Γ_P(x_i, y_i) represents the exit pupil uniformity in the exit pupil position (x_i, y_i), the FOV is divided into M_A × N_A, Γ_A(i, j) represents the uniformity of the angle at the FOV (i, j). The least-squares method is used to optimize the energy distribution of each sub-region in multiple loop iterations to find the minimum value of the constraint errors E_{pupil_error} and E_{angular_error}. The diffraction efficiency corresponding to the specific diffraction order of the grating in each sub-region is then obtained. The propagating energy in the waveguide is redistributed, and the outgoing energy at the 2D exit pupil is uniform to achieve uniformity of the exit pupil illuminance and angular illuminance.

Fig. 5. (a) Exit pupil position sampling regions, (b) FOV sampling points.

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3.2 Analysis of illuminance uniformity

The optimized results of the exit pupil illuminance and angular illuminance distribution are shown in Figs. 6(a) and (b), respectively. The maximum and minimum of the exit pupil illuminance uniformity are 97% and 83%, and the maximum and minimum of angular illuminance uniformity are 72% and 34%. The illuminance uniformity can be evaluated using Eq. (14).

(14)$${\varGamma _\textrm{I}} = 1 - \frac{{{I_{\textrm{max}}} - {I_{\textrm{min}}}}}{{{I_{\textrm{max}}} + {I_{\textrm{min}}}}}$$

where I_max and I_min are the maximum and minimum illuminance distribution values, respectively. After optimizing the illuminance uniformity, the overall uniformity of the exit pupil illuminance can be obtained as 92% at the 2D exit pupil. The overall uniformity of the angular illuminance is 64% in the full FOV.

Fig. 6. Illuminance uniformity distribution, (a) exit pupil uniformity distribution, (b) angular uniformity distribution.

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The diffraction efficiency of each sub-region was obtained by optimizing the illuminance uniformity, which is summarized in Table 2. The above method optimized the diffraction efficiency of each sub-region without considering the structural parameters of the grating, and the diffraction efficiency is the ideal diffraction efficiency of each sub-region. The actual diffraction efficiency of a specific order of the grating varies with different FOV because of the angular selectivity of the grating, which destroys the illuminance uniformity of the waveguide display. To guarantee the consistency of the diffraction efficiency in different FOVs, the structural parameters of the grating should be optimized to improve the uniform distribution of the diffraction efficiency.

Table 2. Diffraction efficiency of each sub-region after optimization

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4. Improvement of the diffraction efficiency uniformity

4.1 PSO optimization method of SSRG diffraction efficiency uniformity

By combining the PSO and RCWA algorithms, an optimization method for the uniformity of the diffraction efficiency of the SSRG at different FOV (including incident angle θ and azimuth angle φ) is proposed, and thus the diffraction efficiency of the grating can be kept relatively constant with the change in the incident FOV. In this process, the ideal diffraction efficiency of each obtained by optimizing the illuminance uniformity is considered as the optimization target diffraction efficiency. The groove depth, fill factor, and slant angle of the SSRG are used as optimization variables. The RCWA is used to optimize the grating structural parameters, so that the actual diffraction efficiency of the different FOV is constrained to continuously approach the target diffraction efficiency during the optimization process. The target diffraction efficiency of the grating in each sub-region is shown in Table 2, and the fitness function of the constrained target is expressed as follows:

(15)$$F = \sum {({{({E_{\textrm{efficiency1}}}(\theta ,\varphi ) - {E_{\textrm{aim - efficiency1}}})}^2}\textrm{ + }{{({E_{\textrm{efficiency2}}}(\theta ,\varphi ) - {E_{\textrm{aim - efficiency2}}})}^2})}$$

E_efficiency1(θ, φ) and E_efficiency2(θ, φ) are the actual diffraction efficiencies of different reflection orders of the grating sub-region under different FOV. E_{aim-efficiency1} and E_{aim-efficiency2} are the target diffraction efficiencies corresponding to each reflection order of the grating. To obtain the minimum value of the fitness function F, the key is how to use the initial point as the starting point to find the next point with the smallest difference from the fitness function.

To improve the efficiency and accuracy of the optimization, it is necessary to develop an efficient optimization algorithm. The PSO algorithm has global search capabilities and is an optimization algorithm for swarm intelligence. This algorithm was first proposed by Kennedy and Eberhart in 1995 and originated from a study of bird predation problems [32]. The PSO algorithm uses massless particles to simulate birds in a flock. Each particle is considered a point in the N-dimensional search space, and its direction is adjusted according to its own and the experiences of other particles. Particles only have two properties: the position of the movement direction and the velocity of the movement. The position is represented as an N-dimensional vector:

(16)$${{\boldsymbol X}_{{\textbf {N,m}}}} = \{ {x_{1,m}},{x_{2,m}},{x_{3,m}},\ldots ,{x_{N,m}}\} ,1 \le m \le {N_{pop}}$$

where X_N,m represents the position in the particle swarm containing N_pop particles and the velocity V_N,m is expressed as an N-dimensional vector:

(17)$${{\boldsymbol V}_{\textbf{N,m}}} = \{ {v_{1,m}},{v_{2,m}},{v_{3,m}},\ldots ,{v_{N,m}}\} ,1 \le m \le {N_{pop}}$$

The PSO algorithm is used in the optimization process. The optimal solutions currently searched by the particle itself and the particle swarm are called the individual optimal and global optimal, respectively. The next particle plans its position and velocity by comparing it with the previous individual and global optima. The particle determines the global optimal solution after multiple iterations. Thus, the optimal solution to the problem is obtained.

This optimization is based on the PSO algorithm combined with RCWA, where the idea is to use the three structural parameters of the grating slant angle α, groove depth h, and filling factor f as the parameters of the optimized particles. The diffraction efficiency of the grating under different FOVs is calculated by calling the RCWA, and the difference between this value and the set target value is used as the criterion for evaluating the quality of the particles. The position of each particle is a three-dimensional vector comprising SSRG structural parameters (α, h, and f) that need to be optimized, and the velocity of the particle is the amount of change in the particle position for each iteration. By continuously adjusting the particle position and speed, the optimal set of particles is finally found in the search range, that is, the current optimal solution, so as to optimize the grating parameters with stable diffraction efficiency.

The process of optimizing the grating structural parameters to improve the stability of the diffraction efficiency using PSO is summarized as follows.

1. Set the range of slant angle α, groove depth h, and filling factor f, and then randomly generate particle position X(α, h, f) and velocity V(δ_α, δ_h, δ_f) according to the set range.
2. Call the RCWA to calculate the best fitness P_best of the individual particle in each iteration and the best fitness G_best of the population among all particles. P_best represents the minimum fitness calculated by a certain particle in the previous iterations and G_best represents the minimum fitness calculated by all particles in the previous iterations.
3. Update the position and velocity of the particles, the update formulas are shown below: $(18)$${\boldsymbol V}_{\textbf{m,k + 1}} = w{\boldsymbol V}_{\textbf{m,k}} + {c_1}rand({{\boldsymbol P}_\textbf{m}} - {{\boldsymbol X}_{\textbf{m,k}}})\textbf{ + }{c_\textbf{2}}rand({\boldsymbol G} - {{\boldsymbol X}_{\textbf{m,k}}})$$$ $(19)$${{\boldsymbol X}_{\textbf{m,k + 1}}} = {{\boldsymbol X}_{\textbf{m,k}}} + {{\boldsymbol V}_{\textbf{m,k + 1}}}$$$ where ω, c₁ and c₂ are weighting factors, and V_m,k and X_m,k are the velocity vector and position vector of the particles in the k-th iteration, respectively. P_m is the position vector corresponding to the historical optimal value P_best, G is the position vector corresponding to the population optimal value G_best, and rand is a random number generated between 0 and 1 with a uniform probability.
4. Judge whether the fitness value F satisfies the conditions of stability and convergence; enter the next cycle or exit to obtain the optimal solution.

If F(X_m,k+1) < F(P_m), then P_m = X_m,k+1. If F(P_m) < F(G), then G = P_m. The changing speed of the particle position is determined by P_best and G_best, and an iteration of the particle position is performed. The iteration stops after G_best satisfies the termination condition and the position vector corresponding to G_best is the optimal grating structural parameter.

According to the above-mentioned PSO algorithm, the diffraction efficiency is optimized under different particle conditions, and the fitness function is calculated. After several iterations, the difference between the actual diffraction efficiency obtained by the current RCWA and the target diffraction efficiency is continuously reduced. When the fitness value is sufficiently small to achieve convergence, the structural parameters of the SSRG reach the optimal value, which can meet the design requirements and guarantee the stability of the grating diffraction efficiency under different FOV. Finally, the illuminance uniformity of the image is realized in the 2D-EPE-SSRG diffractive waveguide.

The entire optimization process is summarized in Fig. 7. Through the non-sequential ray tracing optimization of illuminance uniformity and the PSO optimization of SSRG diffraction efficiency uniformity, the 2D-EPE-SSRG diffractive waveguide achieves high uniformity of exit pupil illuminance and angular illuminance.

Fig. 7. Flowchart of the proposed uniformity optimization method to the 2D-EPE waveguide.

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4.2 Analysis of uniformity distribution of the DOE1 diffraction efficiency

The uniformity of the grating diffraction efficiency can be calculated using following Eq. (20), where η_max and η_min represent the maximum and minimum diffraction efficiencies of the SSRG with different FOV, respectively, and Γ_η represents the uniformity of the grating diffraction efficiency.

(20)$${\varGamma _\eta } = 1 - \frac{{{\eta _{\textrm{max}}} - {\eta _{\textrm{min}}}}}{{{\eta _{\textrm{max}}} + {\eta _{\textrm{min}}}}}$$

The structural parameters of the SSRG in each sub-region were obtained through the PSO optimization method, where the optimized grating parameters and the uniformity distribution of the diffraction efficiency are analyzed. In the waveguide, the R₁ diffraction efficiency of DOE1 is adjusted to be higher than 85% to guarantee that sufficient energy is propagated to DOE2. The decline curve of the fitness function for DOE1 optimized by the PSO algorithm is shown in Fig. 8(a), where the fitness function value after multiple loop iterations is 0.048. The diffraction efficiency distribution of DOE1 for different FOVs is shown in Fig. 8(b). The overall R₁ diffraction efficiency of DOE1 is higher than 85%, which can guarantee sufficient energy propagation. The structural parameters of DOE1 and the uniformity of the R₁ diffraction efficiency are listed in Table 3, where $\bar{\eta }$ represents the average diffraction efficiency with different incident FOV.

Fig. 8. (a) Fitness function decline curve of DOE1, (b) uniformity distribution of DOE1 diffraction efficiency in different FOV.

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Table 3. DOE1 structural parameters and R₁ diffraction efficiency distribution

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4.3 Analysis of uniformity distribution of the DOE2 diffraction efficiency

In DOE2, the grating vector directions of parts A and B are symmetric to the grating vector direction in DOE1. The grating vector values of parts A and B are opposite to each other, and the grating structural parameters of the symmetrical sub-region are the same. As listed in Table 1, in the spherical coordinate system, the incident angles θ of parts A and B are equal, and the azimuth angles φ are opposite to each other. The diffraction efficiency of each sub-region of part A is equal to that of part B, which is symmetrical to it at the FOV (θ, φ). Under the same grating structural parameters, the diffraction efficiency distribution of the sub-regions of part A is the same as that of the symmetrical sub-regions of part B. Here, we list the optimized grating structural parameters and diffraction efficiency uniformity of part A, as shown in Table 4. The diffraction efficiency distribution of grating sub-region 6 is shown in Figs. 9(a) and (b), the diffraction efficiency distribution of grating sub-regions 7 and 8 is shown in Figs. 9(c) and (d), and the diffraction efficiency distribution of grating sub-region 9 is shown in Figs. 9(e) and (f).

Fig. 9. Uniformity distribution of diffraction efficiency in part A of DOE2, (a) R₀ diffraction efficiency distribution of grating in sub-region 6, (b) R_-1 diffraction efficiency distribution of grating in sub-region 6, (c) R₀ diffraction efficiency distribution of grating in sub-regions 7 and 8, (d) R₁ diffraction efficiency distribution of grating in sub-regions 7 and 8, (e) R₀ diffraction efficiency distribution of grating in sub-region 9, (f) R₁ diffraction efficiency distribution of grating in sub-region 9.

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Table 4. Structure parameters and diffraction efficiency uniformity of DOE2

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4.4 Analysis of uniformity distribution of the DOE3 diffraction efficiency

The diffracted light of DOE2 is totally reflected in the waveguide to DOE3 and enters the human eye through the R_-1 diffraction of DOE3. The period of DOE3 is consistent with that of DOE1, and the grating vector direction is the same to ensure that the outgoing FOV is equal to the incident FOV. The grating structural parameters of optimized DOE3 and the uniformity of the diffraction efficiency are listed in Table 5. The diffraction efficiency distribution of the sub-regions 10 and 11 is shown in Fig. 10(a), the diffraction efficiency distribution of sub-region 12 is shown in Fig. 10(b), and the diffraction efficiency distribution of the grating sub-region 13 is shown in Figs. 10(c) and (d).

Fig. 10. Uniformity distribution of diffraction efficiency of DOE3, (a) R_-1 and R₀ diffraction efficiency distribution of grating in sub-regions 10 and 11, (b) R_-1 and R₀ diffraction efficiency distribution of grating in sub-region 12, (c) R₀ diffraction efficiency distribution of grating in sub-region 13, (d) R_-1 diffraction efficiency distribution of grating in sub-region 13.

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Table 5. Structure parameters and diffraction efficiency uniformity of DOE3

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4.5 Analysis of waveguide illuminance uniformity after PSO optimization

The actual diffraction efficiency of the grating in each sub-region was optimized by PSO under different incident FOV. Then, the diffraction efficiency values of the different grating sub-regions with different incident FOV were re-substituted into the 2D-EPE-SSRG diffractive waveguide system, and the exit pupil illuminance uniformity and angular illuminance uniformity optimized by PSO were calculated, as shown in Figs. 11(a) and (b).

Fig. 11. Illuminance uniformity distribution after PSO algorithm optimization, (a) Exit-pupil uniformity distribution, (b) angular uniformity distribution.

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After optimization using the PSO algorithm, the exit pupil illuminance uniformity and angular illuminance uniformity distribution were obtained. The maximum and minimum values of exit pupil illuminance uniformity are 97% and 81%, and the maximum and minimum values of angular illuminance uniformity are 66% and 31%. The illuminance uniformity values can be solved, the overall uniformity of the exit pupil illuminance is 91% at the 2D exit pupil, and the overall uniformity of the angular illuminance is 64% under the full FOV.

5. Projection optics design and waveguide system integration simulation

5.1 Design of the miniature projection optics

A projection optical system needs to be assembled to verify the optical performance of the 2D-EPE-SSRG. The diffractive waveguide substrate is an afocal system that propagates parallel light of the projection system without introducing additional aberrations. A micro-LED microdisplay with a diagonal size of 0.31 inches and a resolution of 1280 × 720 was used as the image source for the projection optics. CODE V optical design software was used to optimize the system parameters to ensure that the system aberration was well corrected. The designed structure of the projection system is shown in Fig. 12(a). The light from the micro-LED is collimated by the lens system and then exits from the projection optics.

Fig. 12. (a) Optical layout of the projection optics, (b) MTF of the projection optics, (c) distortion grid of the projection optics, (d) tolerance analysis of the projection optics.

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The optical modulation transfer function (MTF), distortion, and tolerance of the designed projection optical system are shown in Figs. 12(b−d), respectively. The MTF values of the system are all better than 0.7 at sampling frequency of 30 lp/mm, and the distortion is less than 1%. After manufacturing and assembling, the system MTF values were higher than 0.6 lp/mm. Analysis of the image quality evaluation shows that the projection optical system has good imaging quality and collimation characteristics, which can meet the requirements of the near-eye display system for the eyepiece system. The main performance indicators of the designed projection system are as follows: the focal length is 14.5 mm, the diagonal FOV is 30°, the total system length is approximately 18 mm, and the optical element diameter is less than 8 mm. The projection optical system has a small size and compact structure.

5.2 Simulation of the assembled waveguide system

The projection system was assembled into the waveguide system, and the overall simulation structure of the diffraction waveguide is shown in Fig. 13. To simulate the illuminance uniformity and imaging quality of the image observed by the real human eye, an ideal lens with an aperture of 4 mm was placed at different positions of the 2D exit pupil, as shown in Fig. 14(a). The focal length of the ideal lens was 17 mm, which is equivalent to the focal length of the human eye. A receiver was set on the focal plane of the ideal lens to receive the actual observed image. As shown in Fig. 14(b), four edge positions and one center position were considered as observation points at the 2D exit pupil.

Fig. 13. Integrated simulation of projection optical system and waveguide system.

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Fig. 14. (a) Schematic of the simulated illuminance uniformity of a 4 mm ideal lens at 2D exit pupil, (b) select 5 exit pupil positions for the simulation illuminance uniformity.

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The simulated images of the actual exit pupil positions are shown in Fig. 15, where Fig. 15(a) is the original simulation image loaded on the microdisplay, and Figs. 15(b−f) correspond to the actual images simulated with 1–5 exit pupil positions. Five complete full-FOV images were observed at the exit pupil position with clear imaging and high uniformity image illuminance.

Fig. 15. (a) Original image, (b) simulation of exit pupil position 1, (c) simulation of exit pupil position 2, (d) simulation of exit pupil position 3, (e) simulation of exit pupil position 4, (f) simulation of exit pupil position 5.

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To simulate the actual AR display effect that can be observed by the human eye, a resolution board and real environment were loaded into the designed waveguide system for AR hybrid simulation. As shown in Fig. 16, the human eye can observe that the virtual image is superimposed on the real environment, thereby forming a highly uniform and high-resolution AR display.

Fig. 16. Augmented reality display combined with real environment.

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6. Conclusion

In this paper, a novel optimal design method for a straight-line 2D-EPE-SSRG diffractive waveguide with high uniformity is proposed. To realize the brightness uniformity of the image emitted from the waveguide, the waveguide grating regions are divided, and an illuminance uniformity evaluation model of energy propagation is established. Thereby, the ideal diffraction efficiency of different grating is optimized to realize gradual control, such that the entire exit pupil illuminance and angular illuminance distribution are uniform. However, the actual diffraction efficiency of the grating changes with different FOV, which degrades the illumination uniformity of the waveguide.

On this basis, an optimization method for the uniformity of the diffraction efficiency of the SSRG at different incident and azimuth angles is proposed for the first time by combining the PSO and RCWA algorithms. Three structural parameters of the SSRG are optimized based on PSO to improve the uniform distribution of diffraction efficiency in different FOVs, which further ensures the uniformity of the exit pupil illuminance and angular illuminance of the waveguide. The results of the uniformity analysis and integrated simulation show that the uniformity of the exit pupil illuminance reaches 91% and the angular illuminance uniformity reaches 64%. It has high uniformity in different FOV and different exit pupil positions, and the displayed images are clear. This paper proves that the PSO algorithm can be effectively used for grating design to improve the diffraction efficiency uniformity of a 2D-EPE diffractive waveguide. This new optimized design approach is a promising application for AR-HMDs.

Funding

National Key Research and Development Program of China (2021YFB2802100); Beijing Municipal Natural Science Foundation (1222026); Young Elite Scientist Sponsorship Program by CAST (2019QNRC001).

Acknowledgments

We would like to thank Synopsys for providing the education license of CODE V and LightTools.

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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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Grating	Sub-region	ρ	θ	φ
DOE1	1	0°	-7.9° to 7.9°	-36.0° to 36.0°
DOE2	2	63.4°	-52.6°to -35.1°	0° to 8.4°
	3		-55.8° to -45.5°	-75.3° to -59.8°
	4		-55.8° to -45.5°	-75.3° to -59.8°
	5		-55.8° to -45.5°	-75.3° to -59.8°
	6	-63.4°	-52.6°to -35.1°	-8.4° to 0°
	7		-55.8° to -45.5°	59.8° to 75.3°
	8		-55.8° to -45.5°	59.8° to 75.3°
	9		-55.8° to -45.5°	59.8° to 75.3°
DOE3	10	0°	-52.6° to -35.1°	-8.4° to 8.4°
	11		-52.6° to -35.1°	-8.4° to 8.4°
	12		-52.6° to -35.1°	-8.4° to 8.4°
	13		-52.6° to -35.1°	-8.4° to 8.4°

Grating	Sub-region	Reflection order	Efficiency	Reflection order	Efficiency
DOE1	1	R₁	0.85	R₀	0.25
DOE2	2	R-₁	0.48	R₀	0.52
	3	R₁	0.29	R₀	0.71
	4	R₁	0.30	R₀	0.70
	5	R₁	0.49	R₀	0.51
	6	R-₁	0.48	R₀	0.52
	7	R₁	0.29	R₀	0.71
	8	R₁	0.30	R₀	0.70
	9	R₁	0.49	R₀	0.51
DOE3	10	R-₁	0.06	R₀	0.94
	11	R-₁	0.08	R₀	0.92
	12	R-₁	0.18	R₀	0.72
	13	R-₁	0.48	R₀	0.52

Grating	Sub-region	α (°)	h (nm)	f (%)	η_max (%)	η_min (%)	$\bar{η}$ (%)	Γ_η (%)
DOE2	2,6	28.3	132.9	63.0	η_maxR0= 60.8	η_minR0= 44.7	49.2	84.7
	2,6	28.3	132.9	63.0	η_maxR-1= 54.7	η_minR-1= 40.0	50.4	84.5
	3,4,7,8	56.4	100.3	61.3	η_maxR0= 74.2	η_minR0= 60.2	65.6	89.6
	3,4,7,8	56.4	100.3	61.3	η_maxR1 = 39.4	η_minR1= 25.3	33.9	78.2
	5,9	11.3	99.7	63.5	η_maxR0= 56.3	η_minR0= 45.7	49.4	89.6
	5,9	11.3	99.7	63.5	η_maxR1= 53.9	η_minR1= 43.2	50.2	89.0

Grating	Sub-region	α (°)	h (nm)	f (%)	η_max (%)	η_min (%)	$\bar{η}$ (%)	Γ_η (%)
DOE3	10,11	71.6	629.7	38.4	η_maxR0= 94.3	η_minR0= 86.6	90.8	95.7
	10,11	71.6	629.7	38.4	η_maxR-1= 4.3	η_minR-1= 11.2	7.0	55.5
	12	61.5	107.3	64.8	η_maxR0= 77.6	η_minR0= 70.4	73.1	95.1
	12	61.5	107.3	64.8	η_maxR-1= 23.5	η_minR-1= 17.6	21.2	85.6
	13	63.7	331.7	71.5	η_maxR0= 56.3	η_minR0= 42.5	49.4	86.0
	13	63.7	331.7	71.5	η_maxR-1= 56.2	η_minR-1= 41.8	49.0	85.3

Grating	Sub-region	ρ	θ	φ
DOE1	1	0°	-7.9° to 7.9°	-36.0° to 36.0°
DOE2	2	63.4°	-52.6°to -35.1°	0° to 8.4°
	3		-55.8° to -45.5°	-75.3° to -59.8°
	4		-55.8° to -45.5°	-75.3° to -59.8°
	5		-55.8° to -45.5°	-75.3° to -59.8°
	6	-63.4°	-52.6°to -35.1°	-8.4° to 0°
	7		-55.8° to -45.5°	59.8° to 75.3°
	8		-55.8° to -45.5°	59.8° to 75.3°
	9		-55.8° to -45.5°	59.8° to 75.3°
DOE3	10	0°	-52.6° to -35.1°	-8.4° to 8.4°
	11		-52.6° to -35.1°	-8.4° to 8.4°
	12		-52.6° to -35.1°	-8.4° to 8.4°
	13		-52.6° to -35.1°	-8.4° to 8.4°

Uniformity improvement of two-dimensional surface relief grating waveguide display using particle swarm optimization

Abstract

1. Introduction

2. 2D-EPE waveguide system design

2.1 EPE principles overview

2.2 Initial parameters and profile of diffraction grating couplers

2.3 Principle of the 2D-EPE diffraction

3. Optimization of illuminance distribution in 2D-EPE waveguide

3.1 Optimization of illuminance uniformity

3.2 Analysis of illuminance uniformity

4. Improvement of the diffraction efficiency uniformity

4.1 PSO optimization method of SSRG diffraction efficiency uniformity

4.2 Analysis of uniformity distribution of the DOE1 diffraction efficiency

4.3 Analysis of uniformity distribution of the DOE2 diffraction efficiency

4.4 Analysis of uniformity distribution of the DOE3 diffraction efficiency

4.5 Analysis of waveguide illuminance uniformity after PSO optimization

5. Projection optics design and waveguide system integration simulation

5.1 Design of the miniature projection optics

5.2 Simulation of the assembled waveguide system

6. Conclusion

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (16)

Tables (5)

Equations (20)

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