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Abstract
The optical waveguide is a lightweight and portable scheme for augmented reality near-eye display devices. However, the surface roughness of the waveguide affects its imaging performance, which has not been studied. In this work, we investigate the light scattering caused by the root-mean-square roughness of the waveguide surface and present two methods to numerically analyze the modulation transfer function (MTF) of the display system. Here, we consider the effects of different surface roughness, incident angle, and incident wavelength on the scattering distribution when other conditions are constant. For a simplified optical waveguide display system, the MTF degradation and the variation of the tolerance is calculated. And when the MTF (@ 40 cycles/mm) is required to be 0.3 and the incident angles of the total reflection surface are 45°, 55°, 65° and 75°, the random surface error (RSE) tolerances are 0.207λ0, 0.255λ0, 0.347λ0 and 0.566λ0 (λ0=0.5461µm), respectively. We find a formula descripting the relationship between RSE tolerance and incident angle. If the RSE tolerance exceeds the value of the formula at an angle, the imaging quality of the system will drop significantly. The formula can predict tolerances and incident angles and provide basic tool for imaging quality analysis and manufacturing for optical waveguide AR/VR display systems.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Augmented reality (AR) display is a new display technology that observers can use to observe virtual scenes that overlap in the real world. It is widely used in industrial, military, medical, consumer, entertainment and other fields. The near-eye AR display technology has various solutions, including the free-form prisms scheme [1], the beam splitter scheme [2], the retinal projection scheme [3], the optical waveguide scheme [4–6], and so on. The near-eye AR display system using optical waveguide is light and portable, and is an ideal solution for future AR glasses.
The optical waveguide AR display system consists of a glass substrate and coupling optical elements that direct light into and out of the waveguide. The glass substrate propagates the incoming light based on the principle of total internal reflection (TIR). And the coupling element can be classified into a geometric optical element and a diffractive optical element according to the principle. There are still a lot of challenges of the optical waveguide AR display technology. For the diffractive grating based approaches, the field of view is limited by the waveguide refractive index [7]. And the diffraction efficiency of the optical elements in the display system is not high enough, which influences the performance of the whole system [8,9]. Additionally, a better image quality such as high see-through transparency, high contrast and high luminance is pursued [10]. It's undeniable that manufacturing errors are brought to the glass substrate and the coupling element in the process of processing. One possible factor affecting the image quality of the system is the error of the glass substrate surface. However, there is no work evaluating it in optical waveguide AR display technology.
Surface quality includes surface curvature error, roughness, waviness, and parallelism. The random surface roughness causes the incident light to scatter, which has been studied a lot. The research methods for the scattering of random rough surfaces are divided into the analytical method and the numerical method. The analytical method provides a more intuitive understanding of the characteristics of electromagnetic scattering compared to the numerical method. The most basic and commonly methods in analytical methods are the small perturbation method (SPM) [11] and the Kirchhoff approximation (KA) [12]. Ordinary optical finishing surfaces are micro-rough, and the scattering of light on them needs to be calculated using the Kirchhoff approximation [13].
In this paper, we use the Monte Carlo method to generate a 1D Gaussian random rough surface and use the KA method to calculate the scattered light distribution on the surface, to explain why surface roughness affects image quality. For a simplified optical waveguide display system, we analyze the modulation transfer function (MTF) degradation caused by the roughness of waveguide surface using interferometric (INT) file and tolerancing in CODE V. Moreover, we obtain a formula of the incident angle and the random surface error (RSE) tolerance. This work can provide a foundation for imaging quality analysis and manufacturing error control of the optical waveguide AR/VR display system.
2. Rough surface generation and scattering principle
2.1 Gaussian random rough surface generation
In optical processing, the processed surface will produce random fluctuation. The surface shape of most natural and artificial surfaces can be described by a Gaussian probability density distribution [14], which greatly reduces the computational complexity related to the random process. The Ref. [15] detailly describes the way to establish a 1D Gaussian random rough surface. The reference plane is set in the z = 0 plane of the x-z coordinate system, and the height of the surface relative to the plane is represented by the function z = h(x). The surface roughness is mainly described by statistical parameters root-mean-square (RMS) height ${\delta _h}$ and correlation length ${l_c}$.
The power spectrum function $W(k )$ of a 1D Gaussian rough surface is
where K is the spatial frequency.1D random rough surfaces is generated using Monte Carlo method. A 1D random rough surface with length L composed of N discrete points is expressed as
The generation of a 2D random rough surface is similar to that of a 1D random rough surface [16]. The lengths in the x and y directions are respectively ${L_x}$ and ${L_y}$. And the numbers of discrete points are M and N, respectively. The 2D random rough surface is expressed as
Power spectrum function of 2D Gaussian rough surface $W({K_x},{K_y})$ is described by
2.2 Scattering principle of random rough surface
In this article the KA method is used to research the scattering of 1D random rough surface. And if the RMS slope of the surface is quite small, relatively accurate results can be obtained by using KA single scattering theory alone [15,16]. To simplify the calculation, the shadowing function of the incident light field and the scattered light field are not considered here.
As shown in Fig. 1, the contour height of a 1D rough surface is described by $z^{\prime} = h(x^{\prime})$. The incident light field is ${E_i}(x^{\prime},z^{\prime})$. $\overrightarrow {{k_i}}$ is the wave vector of incident light. $\overrightarrow {{k_s}}$ is the wave vector of scattered light. $\widehat n$ is the external normal unit vector of rough surface. $\overrightarrow {{{r^{\prime}}_s}} (x^{\prime},z^{\prime})$ is the position vector of the scattering point. $\overrightarrow {{\textrm{r}_0}} (x,z)$ is the position vector of the observation point.
The Kirchhoff single-scattered light field of a 1D rough surface is described by
As the RMS slope of the surface increases, the error of the calculation result using Kirchhoff's single scattering theory increases. So, the Kirchhoff multiple scattering model needs to be further established. And the solution of it usually only considers the secondary scattering field [16].
As shown in Fig. 2, after the incident plane light is reflected by the single scattering point, the secondary scattering occurs when the reflected light is incident to another scattering point on the rough surface. The light field after the single scattering is calculated by
$H_0^{(1)}(k{r_{12}})$ is the zero-order first class Hankel function. $H_1^{(1)}(k{r_{12}})$ is the first-order first class Hankel function. And the shadowing function ${S_{12}}$ is described by
3. Numerical calculation and analysis
3.1 Modeling of 1D Gaussian rough surface
To simplify the calculation, we use a 1D Gaussian rough surface as the surface model to study the basic scattering characteristics of random rough surfaces. To satisfy the KA, the radius of curvature of any point on the surface must be much larger than the incident wavelength $\lambda $, which needs to satisfy the following conditions [15]
According to Eqs. (1)- (3), a 1D Gaussian rough surface is produced, as shown in Fig. 3. We set the standard wavelength ${\lambda _0}$ is $0.532\mu m$, the number of sampling points N is ${2^{10}}$, 1D surface length L is $250{\lambda _0}$, and RMS height ${\delta _h}$ equals to ${\lambda _0}$. So, the correlation length ${l_c}$ should be greater than $1.66{\lambda _0}$. The correlation lengths of the surfaces in Fig. 3 are respectively $15{\lambda _0}$, $20{\lambda _0}$, $25{\lambda _0}$, $30{\lambda _0}$. When the RMS height is constant, as the correlation length increases, the surface roughness decreases. Keeping the correlation length constant, reducing the RMS height has the same effect.3.2 Effect of surface roughness on random surface scattering
In order to analyze the scattering distribution of incident light caused by surfaces with different roughness, as shown in Fig. 4, we get the single scattering intensity curve that a 1D Gaussian rough surface with different correlation lengths cause unit amplitude incident light to produce, according to the Eqs. (7) and (8). The intensity is the square of the amplitude $E_s^{[1]}({\theta _s})$.
Fig. 4. Normalized scattering intensity of (a) p-polarized light, when ${l_c}$ is $2{\lambda _0}$, $4{\lambda _0}$, $6{\lambda _0}$, $8{\lambda _0}$, $10{\lambda _0}$, (b) p-polarized light, when ${l_c}$ is $\textrm{15}{\lambda _0}$, $\textrm{20}{\lambda _0}$, $\textrm{30}{\lambda _0}$, $\textrm{40}{\lambda _0}$, $\textrm{50}{\lambda _0}$, (c) s-polarized light, when ${l_c}$ is $2{\lambda _0}$, $4{\lambda _0}$, $6{\lambda _0}$, $8{\lambda _0}$, $10{\lambda _0}$, (d) s-polarized light, when ${l_c}$ is $\textrm{15}{\lambda _0}$, $\textrm{20}{\lambda _0}$, $\textrm{30}{\lambda _0}$, $\textrm{40}{\lambda _0}$, $\textrm{5}0{\lambda _0}$.
For the medium, the refractive index n is 1.57. the RMS height ${\delta _h}$ of surface is ${\lambda _0}$, and the total length L of surface is $100{\lambda _0}$. The incident light waves are s-polarized and p-polarize, the wavelength $\lambda $ is $0.532\mu m$, and the incident angle ${\theta _i}$ is 45°. The distance ${r_0}$ from the rough surface to the scattering observation point is 2 mm, which can be considered as the far field. So, the distance value is considered fixed but the scattering angle is changing. For the intensity curve, we average the results of 500 surface samples. In addition, in order to facilitate the comparison between the scattered light distribution under different roughness, the calculated single scattering intensity in this paper is all normalized by the total scattering energy, i.e. $I_s^{[1]}({\theta _s})/\sum {I_s^{[1]}({\theta _s})\Delta {\theta _s}}$.
Regardless of the polarization state of the incident light, as the correlation length decreases, the scattering range of the outgoing light becomes wider. In the case of large roughness, the peak intensity of the scattered light of p-polarized light is farther away from the surface normal direction than the s-polarized light. In the case of smaller roughness, the difference in scattering intensity distribution between these two kinds of polarized light is negligible. And this is also the case discussed later, where the correlation length is set to $15{\lambda _0}$.
3.3 The effect of wavelength and angle on random surface scattering
If the surface roughness and the incident angle are not changed (${l_c}$ is $15{\lambda _0}$ and ${\theta _i}$ is 45°), the wavelengths of the incident light are changed to 0.488µm, 0.532µm and 0.633µm, and then the obtained single-scattering intensity curve of p-polarized light is shown in Fig. 5(a). The results show that the influence of the incident light wavelength on the scattering intensity distribution can be almost ignored.
Fig. 5. Normalized scattering intensity of p-polarized light, (a) when the wavelength changes and (b) when the incident angle changes.
If the surface roughness and the light wavelength are constant (${l_c}$ is $15{\lambda _0}$ and $\lambda $ is 0.532µm), the angles of the incident light are 45°, 55°, 65° and 75°, and the obtained curve of p-polarized light is shown in Fig. 5(b). The results show that the angles of the peak scattering intensity correspond to the specular reflection direction of the incident angle. But the curves are still different, especially the light emitted at large angles is not considered in the calculation of single scattering. Therefore, the impact of different angles on imaging needs further study.
3.4 Secondary scattering
In order to figure out the importance of secondary scattering, we further calculate the normalized secondary scattering intensity curves $I_s^{[2]}({\theta _s})/(\sum {I_s^{[1]}({\theta _s})\Delta {\theta _s}} + \sum {I_s^{[2]}({\theta _s})\Delta {\theta _s}} )$ at different incident angles according to Eqs. (9)- (12). In the case where the surface correlation length ${l_c}$ is $15{\lambda _0}$ and the incident light wavelength $\lambda $ is 0.532µm, the ratios of the secondary scattering field energy $\sum {I_s^{[2]}({\theta _s})\Delta {\theta _s}}$ to the single scattering field energy $\sum {I_s^{[1]}({\theta _s})\Delta {\theta _s}}$ are as shown in Table 1. The ratios are all less than 2%. Therefore, when calculating the scattering of the precision-machined surface of the optical waveguide, it can be approximated by the Kirchhoff single scattering theory.
Table 1. The ratio of the secondary scattering field energy to the total scattering field energy
4. Simulation and experimental verification
4.1 Influence of surface roughness on MTF of optical waveguide
A basic optical waveguide near-eye display system is shown in Fig. 6. The influence of the roughness of the waveguide surface (TIR surface) on the MTF of the system is studied, when the imaging beam is at different incident angles. The optically coupled elements play a role in the direction of light, which are equivalent to the mirrors. Here, for simplicity, we establish the simplified optical system model in CODE V, replacing the coupled elements with the mirrors and replacing the human eye with the ideal lens. The basic parameters of the model are shown in Table 2.
Table 2. Parameters of optical system model
The article analyzes the effect of TIR surface roughness on the optical system MTF by two methods. One is to attach surface height INT data on the TIR surface, another is to analyze the tolerance of the TIR surface.
For consumer AR head-mounted displays, MTF at 40 cycles/mm is required to be greater than 0.3 [17]. According to Eqs. (4)–(6), 2D Gaussian rough surfaces are generated. We set that the correlation lengths ${l_x}$ and ${l_y}$ are both 0.3 mm, the number of sampling points is $128 \times 128$, the surface size is $1\textrm{8}mm \times 1\textrm{8}mm$, the standard wavelength ${\lambda _0}$ is 0.5461µm, and the RMS heights ${\delta _h}$ are respectively 0.5λ0, 0.55λ0, 0.6λ0 and 1.7λ0. The surface height data is saved in the INT file and attached to the TIR surface in CODE V. We average the MTF curves of 10 height samples for every system, respectively. The results are shown in Fig. 7. When the TIR surface is incident at 45°, 55°, 65° and 75°, and the RMS heights ${\delta _h}$ are the four above, the MTF in the Y direction drops to about 0.3 at 40 cycles/mm. And that in the Y direction degrades more than in the X direction. In addition, we also analyzed that two reflections caused more severe MTF degradation.
Fig. 7. MTF degradation for the height perturbation on the TIR surface when (a) ${\theta _i} = 45^\circ $, ${\delta _h} = 0.5{\lambda _0}$, (b) ${\theta _i} = 55^\circ $, ${\delta _h} = 0.55{\lambda _0}$, (c) ${\theta _i} = 65^\circ $, ${\delta _h} = 0.6{\lambda _0}$, (d) ${\theta _i} = 75^\circ $, ${\delta _h} = 1.7{\lambda _0}$.
In the Tolerancing, the nominal value of MTF at 40 cycles/mm is 0.9549 when the incident angle is 45°, 55°, 65° and 75°. The RSE tolerance of the TIR surface is analyzed choosing MTF at 40 cycles/mm as a performance function. We set correlation distance ${l_c}$ is 0.3 mm and use the inverse sensitivity mode to calculate the RSE tolerance causing the MTF at 40 cycles/mm drop down from the nominal value to 0.3. The RSE tolerance results are 0.207λ0, 0.255λ0, 0.347λ0 and 0.566λ0 (λ0=0.5461µm). If the RSE tolerance values obviously exceed the above results, the imaging performance of the systems will be significantly reduced.
In the same way, the RSE tolerance values corresponding to more angles are calculated. And there is a positive correlation relationship between them, as shown in Fig. 8. This relationship can be fitted to a formula:
where y is the RSE tolerance and x is the incident angle. When the incident angles are 48° and 72°, the fitting results are respectively 0.224λ0 and 0.483λ0, and the results of software calculation are respectively 0.219λ0 and 0.474λ0. The errors, 2.28% and 1.90%, is so small that shows the accuracy of the fitting function. The formula can be used for predicting the further design and fabrication.To verify the relationship between the incident angle on the waveguide surface and the imaging quality of the optical waveguide display system, we fabricate two volume holographic waveguides with same rough waveguide and conduct an experiment. The material of glass is K9, and that of volume holographic grating (VHG) is photopolymer. The image source used is an OLED screen, and the light-emitting wavelength of that is 532 nm. The light is coupled into the waveguide by a VHG and travels in the waveguide by TIR. And the incident angles on the TIR surfaces of the two waveguides are 65° and 75°, respectively. Finally, the another VHG couples the light out and guides it to a camera.
We use a fringe density image to test the imaging quality of the holographic waveguide system. The captured images and the grayscale images are shown in Fig. 9. The MTF in a single frequency is used to evaluate the imaging quality. It is measured by the contrast in the output image assuming the input image is at 100% contrast [18]. And a region with good uniformity (the framed part) is selected as the calculation region. The MTF of the image (a) is 0.4627, and that of the image (b) is 0.5765. The experimental results show that the scattering causes a decrease in the imaging contrast of the optical system. And that proves a larger angle of incidence corresponds to a higher MTF, which means a higher resolution, with the same surface roughness.
Fig. 9. Experimental results. Images (a) and (b) when the incident angles on the waveguide surface is 65° and 75°, respectively. Grayscale images (c) and (d) corresponding to the captured images (a) and (b). Selected calculation region (e) and (f) in images (c) and (d).
4.2 Discussion
The surface of the micro-roughened waveguide after optical finishing not only reflects the light but also scatters it. The direction of the peak scattering intensity is approximately consistent with the specular reflection direction, and but the scattering causes the attenuation of the specular reflectance, which degrades the imaging quality. And the rougher the surface, the wider the scattered light divergence and the lower the energy transfer efficiency of the system. Furthermore, for our subjects, the distribution of scattered light has little to do with the polarization state and the wavelength of the incident light, and the energy of the secondary scattered light is much smaller than that of the single scattered light. So, we could mainly consider the single scattering when numerically calculating and consider the effect of incident angle when simulating. For a simplified optical waveguide display system, the MTF is analyzed using the INT file and the tolerancing. When the MTF degrades, that in the propagating direction degrades more than in the direction perpendicular to the propagating direction. Moreover, within the range of incident angles studied, the larger the incident angle on the TIR surface, the more tolerant the MTF is to roughness. That is, with the same surface roughness, the larger the incident angle, the clearer the system image.
There are some limitations to this research in its present state. Larger incident angles, more reflections and the shadowing function require more complicated calculations, and these can be overcome by further work according to actual needs. In addition, the tilt tolerances of the TIR surface, DLA and DLB, and the curvature tolerance DLF, have little effect on the MTF (@ 40 cycles/mm) within the default tolerance limits of CODE V to be negligible. But for different designed optical systems, a more specific and detailed tolerance analysis is required. And for any system, the formula of the RSE tolerance and the incident angle of the waveguide surface can be derived in the same way as in this paper, which has guiding significance for the selection of the glass substrate, to ensure both image quality and cost.
5. Conclusion
We study the effect of the surface roughness of the glass substrate on the imaging quality of the optical waveguide AR/VR display system. We numerically calculate the scattering distribution and use two methods to analyze the MTF of the system. The rougher the total reflective surface is, the stronger the scattering is, and the more serious the MTF drops. And a formula of the relationship between the RSE tolerance and the angle of incidence is achieved and it could be used for predicting the display imaging quality of the waveguide, as well as indicating that a larger incident angle can allow the surface to be rougher. It provides an evaluating tool for manufacturing and imaging quality analysis of optical waveguide AR/VR display systems.
Funding
National Natural Science Foundation of China (61575024, 61975014).
Disclosures
The authors declare no conflicts of interest.
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