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Abstract
In this paper, we propose a method for using angle offset measurements of an augmented-reality (AR) geometric waveguide partially reflective mirrors array (PRMA), to inform the design and manufacture of future iterations. These offsets are the main cause of display defects such as the formation of double-images. Our design provides a method for quantitative measurement to support the solution of the AR geometric waveguide double-image problem. Our study is based on the principle of optical lever amplification, and PRMA high accuracy measurement is achieved. We also provide an analysis that determines the maximum permissible offset limit of the PRMA geometric waveguide module, without affecting the display quality. The technique is validated on virtual models made in SolidWorks that are synchronized with optical components whose parameters are changed in LightTools. This provides a powerful technique for speeding up the design and manufacturing cycle where until now this has been based on subjective innacurate and unreliable human observation. This paper provides the theoretical basis for the development of a rigorous and reliable measurement technique.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Augmented Reality (AR) [1,2] as a metaverse entrance technology, is highly likely to become the next-generation mobile platform [3] or the next human-machine interface [4]; it is a disruptive technology, which will change the way of human life [5]. In the early stages of development, AR displays with one partially reflective mirror were employed to simultaneously superimpose a virtual image on the real-world scenario [1], an example being using Google Glasses [6]. However, due to its small exit pupil and the small eye box it provided a negative user experience and it never became a mainstream consumer electronic product. The proposed AR geometric waveguide solution replaces one partially reflective mirror with a multiple cascaded partially reflective mirror array (PRMA) to achieve expansion of the exit pupil [7,8]. An example of this type is Lumus [9], which eliminates the negative user experience by having an external shape more like ordinary eyeglass lenses.
AR PRMA waveguides are also called geometric waveguides [10–19], with the principle of simple geometric reflection: as opposed to diffractive and holographic waveguides [20–26] that use light interference. However, the lack of good and stable image display is also an obstacle for AR entering the consumer electronics market. The parallelism of the PRMA components is crucial for good image quality as it induces the defect of a double-image [10,17,27,28] which seriously affects display quality. Unlike traditional optical components, geometric waveguides have all their optical surfaces inside the waveguide, which is not conducive to secondary processing and correction after processing and molding. Due to the lack of a quantitative measurement method for slight PRMA offset, reliance has been placed on subjective human observations and judgement obtained after processing of batch-produced geometric waveguides. Therefore, the parallelism of the PRMA is one of the most important issues in the mass production of waveguides. Even the slightest PRMA offset leads to bad display quality owing to the double-image problem, thereby causing it to be rejected by users.
Motivated by this issue, we therefore propose a measurement method for quantifying the offset. It involves introducing a collimated beam for measuring the AR PRMA waveguide. The system forms each PRMA matching beam spot based on the principle of optical lever amplification. Moreover, the spot position shift characterizes the slight offset of the PRMA, which is then followed by image data acquisition by a charge coupled device (CCD) camera and data analysis. The peak position of the data curve can be used to accurately characterize the spot position, which can be obtained by fitting the data, thereby quantifying the slight PRMA offset. Following this, the conditions for measuring beam coupling-in the waveguide, the PRMA offset relationship with beam offset and field of view (FOV). The offset limit of the PRMA angle for the geometric waveguide module was also obtained without noticeably compromising the display performance. Also, the simulation results and measurement error analysis are elaborated.
2. Measurement system design
As shown in Fig. 1, there are four main components in our design, viz. the collimated light source system, the geometric waveguide to be measured, the spot receiver plane and the CCD device for data acquisition.
Total internal reflection (TIR) transmission is performed in the front part of the geometric waveguide after the collimated light source leading to the coupling-in wedge surface of the geometric waveguide to be measured. The wedge surface angle of the coupling-in side should be set to a value such that the vertical incidence measurement beam corresponds to the central FOV. This can also be achieved, by additional processing of the matching triangular prisms without changing the original wedge surface angle, thus meeting the actual production needs. Subsequently, the beam is coupled into the geometric waveguide and transmitted by TIR to the 1st PRMA. The distance from the coupling-in side to the 1st PRMA is usually determined based on the distance from the user’s eye pupil to the outer corner of the eye. After the TIR beam reaches the PRMA, part-transmission and part-reflection occurs; where the transmitted part is propagated to the next PRMA behind it and the reflected part is coupled out of the geometric waveguide.
When the highly parallel PRMA meets the design requirements, the coupled-out measurement beams will also be highly parallel. When there is a slight offset in the PRMA, the corresponding coupling-out beam undergoes a matching angle shift. The measurement beam that is coupled out from the waveguide is incident on the distant receiver plane. Based on the principle of optical lever amplification, a slight PRMA angle offset can cause the distant beam spot to shift by a significant amount. The corresponding coupling-out beam is incident on the receiver plane at the corresponding spot, and the data is captured and acquired by the CCD camera. According to the position data of the spot acquired by the CCD for Gaussian fitting to determine the central position of the spot, the CCD can be selected to identify the accuracy at a resolution of 0.01 mm, thus completing the calculation of the slight offset of the PRMA.
2.1 Measurement beam coupled into waveguide
Figure 2 shows that there are five PRMA surfaces inside the AR geometric waveguide, and that the regions with light coupling-out from the PRMA belong to the FOV area, where the center of the 3rd PRMA element is at the central FOV position. When performing the measurements in our study, the measurement beam was not coupled in with the geometric waveguide from the middle of the right-side wedge surface. Instead, the measurement beam was required to be coupled-in with the geometric waveguide from a specific position on the wedge surface, such that the coupling-out beam corresponds to the central position of the FOV.
Fig. 2. The measurement beam coupling-in wedge surface of geometric waveguide and coupling-out from the FOV area.
As shown in Fig. 2, the inclination angle of the 5 PRMAs of the AR geometric waveguide is $\theta {_{PRMA}}$ , the thickness of the geometric waveguide is $d$, and the distance between the center FOV position and coupling-in side is $l_{center}$. The measurement beam is transmitted in the geometric waveguide with TIR, and when the PRMA is encountered, the reflection part is coupled out of the geometric waveguide. The angle $(\alpha )$ between the waveguide surface and TIR beam of the geometric waveguide is calculated as follows:
Here, the length of the beam that returns to the ipsilateral waveguide upper surface after one TIR transmission within the geometric waveguide is defined with a period of $P_0$
From the coupling-in to the coupling-out position, the beam can have N periods, each of length $P_0$ . Meanwhile, the beam coupling-in side and the central FOV coupling-out side represent incomplete period lengths, where the length of the coupling-out position is $P_{out}$ , and that of the coupling-in side is $P_{in}$.
In this case, the position of the measurement beam on the wedge surface of the coupling-in side can be indicated as $l_{in}$.
Position $l_{in}$ corresponds to a FOV of 0$^{\circ }$ at the center of the FOV. It can be seen that $l_{center}$ is satisfied by fulfilling the following condition:
Although it is preferred for the measurement beam to enter at position $l_{in}$ on the wedge surface, it is not mandatory. It is worth pointing out that in our study, it is required that the tilt angle $\theta _{couple-in}$ satisfies the following relationship:
We can cut the wedge surface of the geometric waveguide to satisfy this angle relationship. However, in an actual industrial geometric waveguide product, usually the wedge surface angle does not satisfy this relationship to meet the matching front projection module system in a convenient way. This issue can be solved by gluing a matching trigonal prism to the wedge surface, such that it meets the measurement beam entry requirements.
2.2 PRMA offset relationship with FOV
As shown in Fig. 3, the virtual image can be viewed through the AR geometric waveguide. Its size can be tens or hundreds of inches. The central position is the optical axis of the central FOV. Usually, diagonal FOV is one of the core parameters of AR display module. The FOVs in the horizontal and vertical directions can be defined as $FOV_{Horizontal}$ and $FOV_{Vertical}$, respectively.
Fig. 3. Schematic of various FOVs of virtual image observed by eyes through geometric waveguide module.
For a given parameter of the AR geometric waveguide display module, its vertical and horizontal FOVs depend on the aspect ratio of the microdisplay used in the module. As shown in Fig. 4, the aspect ratio of the FOV, given by, W:H, can be determined on the basis of the aspect ratio of the microdisplay. Then the vertical and horizontal FOVs can be calculated in terms of the diagonal FOV as follows:
Taking the industry’s mainstream geometry waveguide with a FOV of 40$^{\circ }$ as an example, the microdisplay aspect ratio is 16:9. Thus, $FOV_{Horizontal}$ and $FOV_{Vertical}$ can be calculated as 34.9$^{\circ }$ and 19.6$^{\circ }$, respectively.
If the central FOV is defined as 0$^{\circ }$, the horizontal FOV can also be correspondingly defined as $\pm$17.45$^{\circ }$. As shown in Fig. 5, the upper and lower lines represent the upper and lower surfaces of the AR geometric waveguide, respectively, while the middle slash represents the PRMA of the geometric waveguide. The AR geometric waveguide material is H-K9L, having a refractive index of $n_{d}$=1.51673. It can be seen that the horizontal FOV inside the geometric waveguide is smaller than that outside the waveguide
The FOV corresponding to a single pixel $(FOV_{per-pixel})$ can be simultaneously obtained. Taking the AR geometric waveguide module of the actual microdisplay with 1080P resolution as an example, the pixel count is 1920 in the horizontal direction. The FOV range corresponding to a single pixel is given as $FOV_{inner per-pixel}$.
Meanwhile, the angle range corresponding to a single pixel within the geometric waveguide is $FOV_{inner per-pixel}$.
2.3 PRMA offset relationship with a measurement beam offset
Unlike the AR display module with a single half-transparent and half-reflective surface, the virtual image is displayed by only one surface. For AR geometric waveguide displays, the virtual image is formed by combining multiple PRMAs. If all PRMA surfaces are strictly parallel to each other, the displayed image will be excellent. If there is a slight angle offset in one of the PRMAs, it will induce a double-image effect on the display.
As shown in Fig. 6(a), the blue line represents the PRMA, and incident angle is strictly equal to the reflection angle when the measurement beam is transmitted by TIR inside the geometric waveguide
Fig. 6. TIR measurement beam reflected on PRMA inside the geometric waveguide. (a) correct PRMA position and (b) offset PRMA position.
When a slight angle offset of PRMA occurs, it causes a shift of the reflected beam. As shown in Fig. 6(b), the blue line indicates the location of the offset PRMA, while the red dotted line indicates the correct PRMA position. PRMA offset angle is indicated by $\theta _{PRMA-offset}$.
Notably, the $\theta _{PRMA-offset}$ offset of PRMA causes a measurement beam angle shift of $2\theta _{PRMA-offset}$.
The non-parallelism of the upper and lower surfaces of the geometric waveguide can have an impact on the AR geometric waveguide by creating double-imaging. However, it is not clear that this will be an influencing factor in the actual process of manufacturing AR geometric waveguides for mass production. During the analysis of the theoretical model of the actual AR geometric waveguide display module, light is first transmitted by TIR within the geometric waveguide, which passes over the upper and lower surfaces of the geometric waveguide. After TIR in the geometric waveguide and then reflection off the PRMA, it is then coupled out.
Therefore, the illusion from the first subjective impression appears to indicate that the parallelism of the upper and lower surfaces of the geometric waveguide is one of the factors causing the double-imaging of the geometric waveguide. However, in the geometric waveguide, the upper and lower surfaces of which belong to the outer surface of common optical components are produced using an already mature technology where parallelism is determined by well-established inspection techniques that ensure that they meet AR geometric waveguide imaging requirements.
Also, the parallelism of the upper and lower surfaces of the geometric waveguide can be corrected by secondary processing to make it meet the geometric waveguide requirements; this is also described later in Section 3.1. Therefore, the factor affecting geometric waveguide double-imaging can be solved in the actual industrial manufacturing process and does not impact upon the mass production of geometric waveguides. As there are mature technologies to solve this issue, it is not included as a consideration point in this paper.
2.4 PRMA angle offset limit
The highest resolution limit of the display image for the AR geometric waveguide module is a single pixel of the microdisplay. In this case, as long as the angle offset of the PRMA induces a double-image effect of less than one pixel on the actual image, it is unnoticeable. The relationship between the PRMA offset and the shift of the measurement beam is as follows:
As already deduced above, the offset angle of a single pixel-width ray is equal to the FOV range corresponding to a single pixel. The PRMA offset for a single pixel range is as follows:
It can be calculated for a geometric waveguide module with 1080P resolution, FOV = 40$^{\circ }$, and H-K9L glass material, with the following specifications: $W_{pixel-number}$=1920, $FOV_{Diagnoal}$=40$^{\circ }$, $n_d$=1.51673. The angle that allows the maximum PRMA offset is 21$^{\prime \prime }$ i.e., less than this angle the image display quality is not affected.
The allowed PRMA offset limit is expressed as a function of $FOV_{Diagnoal}$, $n_d$, and $W_{pixel-number}$.
The larger the FOV of the geometric waveguide display module, the smaller is the PRMA angle offset limit. The larger the refractive index of geometric waveguide, the smaller is the PRMA angle offset limit. The higher the resolution of the microdisplay used in the module, the lower is the PRMA angle offset limit.
As shown in Fig. 7, the graph indicates the relationship between $n_d$ value of the geometric waveguide glass material, where the red and blue lines indicate the geometric waveguide module of FOV 40$^{\circ }$ with resolutions of 1080P and 720P, respectively. For a AR geometric waveguide display module of 40$^{\circ }$ FOV made of H-K9L glass material, the maximum PRMA angle offset limits are 21$^{\prime \prime }$at 1080P resolution and 32$^{\prime \prime }$ at 720P resolution. If the $n_d$ of the geometric waveguide material is increased to 1.8 when using ZF7L glass material, the PRMA angle offset limits change to 18$^{\prime \prime }$ at 1080P resolution and 27$^{\prime \prime }$ at 720P resolution.
Fig. 7. Changes of maximum PRMA angle offset limits for 1080P resolution and 720P resolution with refractive index of the geometric waveguide various materials.
The figure shows the data for different optical glass materials in red font, and the data for common optical plastics, such as PMMA, PC, and injection molding optical plastics commonly used in cell phone cameras in blue font. The relevant $n_d$ values for the different optical plastics and optical glasses are shown in Table 1.
Table 1. Refractive index of various optical glasses and plastics.
The optical glass in the Table 2 is the CDGM glass catalog type [29], and the corresponding SCHOTT glass catalog type [30] in brackets, where CDGM’s H-K9L corresponds to SCHOTT’s N-Bk7, and their parameters are slightly different. All relevant data are referenced from its official website.
Table 2. Parameters used for the proposed measurement method.
As can be seen from the table, as the $n_d$ value of the optical glass gradually increases, the allowed angle offset corresponding to PRMA becomes stricter. The AR geometric waveguide is a new type of optical element, in which all the optical surfaces are planar and which also does not introduce additional optical aberrations. In this case, for the same front projection module, replacing different geometric waveguide glass materials will improve the FOV of the AR display module. If we take the front projection system comprising a geometric waveguide made using H-K9L glass material with an FOV of 40$^{\circ }$ as an example, varying the refractive index of the material, the diagonal FOV can be expressed as follows:
The FOV is positively correlated with the $n_d$ of the glass material, and the higher the $n_d$ value of the glass material of the geometric waveguide, the larger is the FOV.
As shown in Fig. 8, the horizontal axis represents the different refractive indices and the vertical axis shows the FOVs. The projection system is the front projection module of the H-K9L glass geometric waveguide with an FOV of 40$^{\circ }$. The FOV of the display module can be increased to 48$^{\circ }$ when the $n_d$ of the geometric waveguide is increased to 1.8 by using the ZF7L material.
Fig. 8. Changes of diagonal field-of-view with refractive index of the geometric waveguide various materials.
Although ideally, the use of a glass having a refractive index as high as 1.8 would provide good performance, in practice this cannot be realized so we chose a glass with a more realistic refractive index of 1.51. If use a glass having a refractive index of 1.8 for the geometric waveguide, we can raise the FOV of the AR module to 48 degrees, and under the same conditions, we can raise the FOV by at least 20% as indicated by our results in Fig. 8.
Exactly, this is limited to the effect of the current availability of glue refractive indices. Therefore, currently in the AR geometry waveguide manufacturing industry, the glass material for AR geometry waveguide is not freely chosen at will. Rather, the refractive index of the glass substrate material is selected based on the refractive index of the available glue. So with the limited available refractive index of the glue, we sacrifice the glass material to match the need to meet the refractive index of the glue.
As the availability of a suitable glue is the current limiting factor, a possible solution is to avoid the use of glue altogether. A process that can use glass bonding is being actively explored within the AR geometric waveguide manufacturing industry to implement the geometric waveguide gluing process without glue. It is an alternative to glue by achieving glass bonding without an intermediate glue layer [31]. In Figs. 7 and 8, we show a number of higher refractive index materials as alternatives for geometric waveguide glass substrates and show their performance in terms of FOV in degrees. We consider that glue-less glass bonding is an important direction in the development of AR geometric waveguides over the coming years that will significantly improve the important parameter of FOV.
Notably, glasses with different $n_d$ values differ in other properties, e.g. the scratch resistance of some glass materials may be reduced at high $n_d$ values. At this point, the performance requirements of the consumer electronics display modules include resistance to scratches, etc. Therefore, coating the surface of glass materials with a functional film that resists scratching may be a good direction to look into. Moreover, it would be wise to research glass materials that fulfil the requirements of consumer electronics display modules.
2.5 Measuring PRMA angle offset
The AR geometric waveguide optical surfaces are located inside the optical element, which is different from conventional optical elements. This leads to the development of a different method for measuring the parallelism of its PRMAs. The method of incident measurement beams from the wedge surface of the geometric waveguide is a very good solution. The displacement of the spot position $(\Delta x)$ on the distant receiver plane is caused when the beam is shifted by $\theta _{beam-offset}$, when the PRMA has a slight offset.
where $l$ is the distance from the receiver plane to the geometric waveguide. At the same time, the beam inside the geometric waveguide offset s by $\theta _{inner-beam-offset}$.Meanwhile, the beam shift angle shows a two-fold relationship with the offset of PRMA, which can be indicated as follows:
In other words, based on the principle of optical lever amplification, when the distance between the receiver plane and the geometric waveguide is changed, the PRMA offset can be measured with high accuracy. When the distance requirement is large enough, PRMA offset measurement can be performed in the form of a reflector-folded optical path with a mirror.
Equation (23) gives the PRMA offset based on the displacement of the spot position (${\Delta x}$) that has been measured and has a quantified value. However, there are various reasons why we did not use the displacement of the spot position (${\Delta x}$) to evaluate the PRMA of waveguide. First, the main cause of the AR geometry waveguide double-imaging is due to the offset of the PRMA, which results in inconsistent images formed on each PRMA surface, and produces double-imaging. Therefore, according to the intuitive reason of double-imaging, defining it by the offset of PRMA is the optimal way to define the evaluation.
Second, in our quantitative measurement research work, the spot offset corresponds to the offset of the PRMA. Therefore, using the displacement of the spot position to evaluate the AR geometric waveguide is only applicable to our proposed measurement scheme and can be used in this way. With the further development of research and follow-up work in the AR industry, a variety of methods may be proposed for measuring geometric waveguide double-imaging. At this point, if another researcher follows-up with a new method to measure the double-imaging of the AR geometry waveguide, it may not use the quantitative information of the spot shift. Therefore, using the spot shift information to evaluate the AR geometric waveguide is only applicable to our scheme and is not universal. On the other hand, the use of PRMA offset to define the double-imaging of geometric waveguides is generalizable.
Third, our study uses the principle of light lever amplification, so the the spot shift is not fixed and will change with the distance of the detection receiver plane. Therefore, it is clearly inappropriate to use an uncertain amount of spot shift to evaluate a definite AR geometric waveguide double-imaging.
3. Results and discussion
3.1 Simulation
In this study, we used SolidWorks [32] 3D structural software and LightTools [33] optical software for simulation. First, the mechanical structure of the AR geometry waveguide and other related optical components were simulated in SolidWorks, and then the related 3D mechanical structure files were imported and linked to the LightTools optical simulation software. The PRMA offset of the geometry waveguide was also set in SolidWorks. One of the advantages was that changing the structural parameters in SolidWorks can be synchronized with changing the optical component parameters in LightTools.
The optical simulation is shown in Fig. 9, which consists of the right-hand side of the collimated light source system, and the outgoing collimated measurement beam coupling-in the geometric waveguide wedge surface into the waveguide in TIR. There are three PRMAs in the AR geometry waveguide; the second PRMA has a slight offset of 3$^{\prime \prime }$. All three PRMAs were set with the corresponding transmittance and reflectance, and appropriate parameter values were chosen to ensure good overall transmittance of the waveguide and uniformity of the display plane brightness. The measurement beam reflected by the PRMA was coupled out of the geometric waveguide to the distant receiver plane, thus forming the measured spot. The spot position data of the receiver plane was analyzed and the offset of the corresponding PRMA was calculated from the measurement.
In the simulation, the relevant optical characteristics of each optical element, including the reflectivity and transmissivity of the PRMAs, were set, along with the parameters for the glass material and the collimated light source. The relevant parameter settings are shown in Table 2.
where $\Phi _{beam}$ is the measurement beam aperture, $T_{waveguide}$ is the overall transmissivity of the geometric waveguide, and $T_{1st-PRMA}$, $T_{2nd-PRMA}$, $T_{3rd-PRMA}$ are the transmissivities of the 1st, 2nd, and 3rd PRMA.
Notably, the parallelism of the upper and lower surfaces of the geometric waveguide can also introduce defective display effects. However, the parallelism of the upper and lower surfaces can be processed relatively easily and enables secondary process corrections that are readily measured. On the other hand, the dimensional processing accuracy of glass elements is now easily higher than 0.01 mm, and the lateral shift of PRMA can also be easily identified. For a small lateral shift (0.01 mm), the percentage of its influence on the angle offset of PRMA measurement is negligible.
According to the parameters set above, a spot offset of 0.44(1) mm, as shown in Fig. 10, and a PRMA offset of 2.99$^{\prime \prime }$ are obtained.
In Table 2 it should be noted that the path length ${l}$ is created with the use of a mirror that forms a folded optical path. The measurement beam undergoes reflection from one mirror that is located at half distance. In this way, a sufficiently long light path can be created in the limited size of the laboratory or factory floor.
Optimizing the PRMA offset depends on the structural parameters inputted into SolidWorks and changing the optical component parameters in the LightTools simulation. The offset of the PRMA is simulated by changing the geometric waveguide structure parameters by small amounts. When the PRMA is slightly shifted, it causes the defect of AR display module double-imaging display, which is one of the important factors plaguing the mass production of AR geometric waveguide consumer electronics. Double-imaging defects in the AR display module are one of the reasons for the low yield in the actual manufacturing process of the geometric waveguide.
AR geometric waveguides are distinguished from conventional optical elements as a new type of optical element, so we must devise a means of simulating these. With conventional optical elements, the optical surface is on the outer surface, such as the left and right side of a lens. For conventional optical components, secondary machining can be used when the machining parameters do not meet the design requirements.
However, we need to consider whether this approach works for optimizing AR geometric waveguides where the optical surfaces are located inside the device as in this case it is not possible to optimize the PRMA offset in the form of a secondary processing correction. At this point, the current study provides quantized AR geometric waveguide measurements that give quantified values of the PRMA offset. The waveguide process engineer can improve the process parameters of the next batch of AR geometry waveguides based on the PRMA offset quantization values of the current batch; the production of an improved product is obtained through an iterative process.
For an AR near-eye display system, image uniformity is one of the important core parameters of the AR display module. For geometric waveguide AR display modules, different reflectance of each PRMA surface is usually used to ensure the uniformity of the AR display image. The light is transmitted by TIR within the geometric waveguide, and the reflectivity of the front PRMA surface is low due to the high intensity light received by the front surface. As the PRMA surface moves backward, the light energy couples out of the waveguide into the human eye. The intensity of the light transmitted by TIR within the waveguide decreases with distance, so the subsequent PRMA reflectivity gradually increases to compensate for this. In order to ensure the uniformity of the image, the reflectivity of each PRMA surface should satisfy the following relationship [34]:
where $R$ denotes the reflectivity of a PRMA surface and $i$ denotes the number of PRMA. The first PRMA is near the geometric waveguide coupling-in side, and the sequential numbers are arranged from it. If the transmittance is not considered, the reflectivity of each PRMA is 1/5, 1/4, 1/3, 1/2, 1 in order. However, the AR waveguide is a see through display and can see the external environment at the same time, so the last piece of PRMA cannot be set as opaque.As mentioned in Table 2, the AR display module transmittance parameter is 85%, so the last PRMA reflectance is usually set to 15%, that is, $R_5$=15%. According to Equation (24), $R_4$=13.0%, $R_3$=11.5%, $R_2$=10.3%, and $R_1$=9.4% can be calculated sequentially. In this paper, we study a measurement scheme where only a single beam is transmitted by TIR within a geometric waveguide. In this case, the PRMA coating reflectivity is the main cause of the coupling-out spot uniformity. The definition of uniformity of the AR display image is given in the previous work of C. P. Chen et al. [35] as follows:
For example, when the absolute error of PRMA reflectivity is $\pm$3% in the coating process, to calculate the spot image uniformity. At this point, the laser beam used for our measurements is fine enough that the individual spots do not overlap, the corresponding image uniformity is 68% caused by the coating process. The image uniformity of the geometric waveguide was reported to be less than 70% in the study of the article by L. Gu et al. [34]. The actual AR geometric waveguide module image uniformity is related to the uniformity of the microdisplay device itself and the projector optical system design. This is only a brief illustration of the results of the single-beam measurement of the spot uniformity in our scheme. Because the measurement scheme in this paper is based on the position of the spot, there is no direct relationship with the intensity of the spot.
3.2 Measurement error analysis
As shown above, (23) is the formula for the PRMA measurement of this study. The PRMA angle offsets are associated with the $n_d$ value of the geometric waveguide glass material, the $\Delta x$ value, and the distance $(l)$ between the geometric waveguide and the receiver plane. In other words, the geometric waveguide PRMA offset is a function of the three parameters, as follows:
The accuracies of $\Delta x$ and $l$ are $\pm$0.01 mm and $\pm$1 cm, respectively, while the $n_d$ value of the AR geometric waveguide for a specific material can be considered a constant. The measurement accuracy of the PRMA offset can be calculated as follows:
The absolute accuracy of the PRMA offset can be calculated with a measurement accuracy value less than 0.07$^{\prime \prime }$.
The display double-image problem caused by the PRMA position offset puts a major constraint on the mass production of geometric waveguides in the AR industry. Earlier, operators used to observe a series of geometric waveguide modules by eye and select those with good display effects without the double-image issue. However, it is difficult to find a measurement method to quantify the double-image problem of geometric waveguides.
As shown in Fig. 11, for the virtual display plane of resolution 1080P, the width of the strokes of the subtitles is usually about 2-4 pixels, which people observe and judge the display effect form batch-produced geometric waveguides. For an AR geometric waveguide display module with a FOV of 40$^{\circ }$, 2 pixels of double-image correspond to a PRMA offset in the range of 43$^{\prime \prime }$. Thus, the measurement method proposed herein greatly improves the measurement accuracy of PRMA to 0.07$^{\prime \prime }$ as well as the quantified values.
Fig. 10. Simulated measured spot shift caused by PRMA offset. Spot shift $(\Delta x)$=0.44(1) mm, consistent with 2.99$^{\prime \prime }$ offset of PRMA, which agrees with the foregoing results.
Fig. 11. The double-image defects of the display that can be observed by the human eye with reference to the stroke width in an image of resolution 1080P.
The offset value of 43$^{\prime \prime }$ PRMA offset is specific to the particular parameters of 40$^{\circ }$ FOV and AR display module microdisplay resolution of 1080P. This corresponds to a PRMA offset of 2 pixel points, which is the width of the strokes of the movie subtitles. The procedure for calculating the value of 43$^{\prime \prime }$ is shown below, which corresponds to 2 pixels offset and is a variation of the earlier Equation (18):
With the relevant known parameters, the 43$^{\prime \prime }$ PRMA offset corresponding to the 2 pixel double-imaging can be calculated.
4. Conclusion
A quantifiable, high-precision method for measuring PRMA offset is proposed herein, which can replace the previously reported method of judging the geometric waveguide double-image subjectively with the human eye. It also provides an accurate method for measuring mass production in the AR geometric waveguide industry. The measurement method is easy to operate, robust and accurate up to 0.07$^{\prime \prime }$. The maximum allowable PRMA angle offset of AR display modules with different refractive indices and different resolutions are also given in this study. Our future research will extend to quantifying other core parameters of geometric waveguides, such as brightness and uniformity. Although only simulations have been carried out at this stage, the work has paved the way for practical measurements to be made for expediting the progress between successive iterations of design and fabrication of geometrical waveguides for AR applications.
Funding
National Natural Science Foundation of China (12004261, 12174264, 61775142, 62075138, 92050203); Key-Area Research and Development Program of Guangdong Province (2019B010925001).
Acknowledgments
Key-Area Research and Development Program of Guangdong Province (Project No. 2019B010925001).
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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