Freeform prism systems are often used for head mounted display systems for augmented, virtual, and mixed reality as they have very few independently moving elements while also having a large field of view (FOV) in a small and light form factor [1-3]. With relatively few available variables, but a large eyebox and FOV, these systems are often difficult to achieve the required optical performance. This is because they are composed of a single material, typically polymethyl methacrylate (PMMA), and they have no symmetry around the pupil or image. With the use of a single material and lack of symmetry, the aberrations due to color are often significant and unavoidable. This paper attempts to mitigate one of these issues, the single material, by adding an additional variable to the system by allowing the index of refraction profile of the prism to vary, which is known as a gradient-index (GRIN) material.
With the implementation of GRIN there is the potential to reduce the lateral color with the added degree of freedom. It has been shown that a PMMA/Polystyrene plastic GRIN can be used to correct residual lateral color in an eyepiece while maintaining the same weight as homogeneous PMMA . This is achieved by the fact that GRIN materials allow for light to bend within the material instead, so that the dispersion due to the gradient of the material can be equivalent to some materials that do not exist on the normal glass map. Without the GRIN, the rays in these designs are mostly reflecting off the surface of the material, and only the dispersion property of one material contributes to the chromatic aberrations.
Gradient index optical materials have a refractive index variation throughout the lens volume. The index distribution can in theory take on arbitrary form, but there are three common geometries, axial, radial, and spherical, shown in Figure 1-1. The isoindicial surfaces (contours of constant index) are planes, coaxial cylinders, and concentric spheres respectively . The axial gradient with planar surfaces of constant index can be used to correct spherical aberration, similar to a homogeneous aspheric surface . On the other hand, radial and spherical GRIN profiles are used more often in broadband lens designs where chromatic aberrations are usually the limiting aberrations. There is an additional benefit in the radial and spherical geometry as the index changes perpendicular to the optical axis, causing the marginal ray to bend, introducing optical power. In a homogeneous lens, the optical power comes from the ray bending at the surfaces of the lens. But with a radial or spherical GRIN lens, along with the surface power, there is bending inside of the lens, which adds additional power and another degree of freedom to correct chromatic aberrations.
The axial GRIN geometry is typically the least difficult to manufacture. But since its primary benefit is approximately the same as aspheric surfaces, which are usually more economical to manufacture, the use of axial GRIN is greatly limited. However, axial GRIN can be a precursor for spherical GRIN and other GRIN distributions by compression and reshaping [4, 7]. One of the most outstanding optical properties of GRIN materials is that they can have unique dispersion properties that do not exist in the currently discovered homogeneous materials. The Abbe number of a GRIN material is defined as,
where Δn is the change of index of refraction at the short, center, and long wavelengths . This is defined by its power contribution to different wavelengths, which can theoretically be as small as zero, infinity, or even negative. This ability to have unique dispersion properties is critical to optical designers for reducing the number of elements in a design and decreasing the overall weight and complexity of broadband systems.
Additive manufacturing of optical lenses
The ability to create GRIN geometries more complicated than the traditional axial, radial, and spherical are coming about due to the improved development of additive manufacturing. The technology of additive manufacturing, also known as 3D printing, has increased to the point that optical quality lenses can be 3D printed [9, 10]. These lenses have the potential to reduce the amount of waste created by the traditional grinding of lenses, which leave small plastic residue which pollutes the water and earth, by not having to grind at all. The lenses can be printed with optical quality to the specified shape without the need to grind or polish. The other advantage of these lenses is that they can be processed on the spot right as someone is ordering their glasses. Instead of having to wait several days for their lenses, the lenses can be printed in a matter of hours. Another potential of 3D printed lenses is the ability to print unique prescriptions that are much more difficult to keep either stocked or even to produce.
The improved development of additive manufacturing for optical lenses also brings up the possibility of 3D printing of GRIN lenses. If the system can print homogeneous lenses, the system can also print GRIN lenses. Normally for homogeneous lenses, the goal is to get rid of any changes of index within the processing where any sort of changes would be considered defects. But the goal of GRIN lenses is to be able to tailor the index change. There has been an increasing amount of development in 3D printing of GRIN lenses [11-13]. This technology can be used for creating small lens arrays with flat surfaces, to even larger monolithic prisms with unique profiles that would otherwise be impossible to make. This study will look at the use of a 3D printable GRIN for the improvement of performance in monolithic MR systems.
The specifications for the design are shown in Table 1-1. These specifications are derived from the minimum requirements for the system to be able to have an angular pixel resolution of 1.5 arc minutes. In order to have the required angular pixel resolution, with the focal length of 21 mm, the performance needs to be at 50 lp/mm instead of 30 lp/mm as given in the abstract and competition. The other specifications are there to push the system to have a large FOV within a small weight and size.
Specifications for homogeneous and GRIN freeform reflective prism.
|Eye Clearance (mm)||> 18.25|
|Wavelengths (nm)||486, 587, 656 (d, F, C)|
|FOV (deg.)||26 × 41 (H × V)|
|Distortion (%)||˂ 12|
|Image Quality||MTF > 10% @ 50 lp/mm|
|Optical Path Folding Direction||XZ Plane Symmetry|
|Number of Freeform Surfaces||4|
|Total Number of Freeform Coefficients||15|
|Number of Freeform Surfaces||4|
|Total Number of Freeform Coefficients||15|
|Diagonal Length of Active Display (mm)||218|
|Resolution in Pixels||1080 × 1920 (X × Y)|
|Pixel Pitch (μm)||10|
HOMOGENOUS STARTING DESIGN
A homogeneous monolithic design with freeform surfaces is used as a baseline comparison for the GRIN design. This design does not employ a total internal reflection (TIR) surface that is typically seen in this application. Instead this design study uses the geometry referred to as the reflective prism . This is because this geometry seems more ideal for adding a GRIN material as the path through the system is more direct. Also, Takaki, et al. reported a study of the TIR versus reflective prism in terms of their freeform aberration correction . Takaki, et al. discovered that the desire to correct color aberrations in the TIR prism, instead of focusing of the freeform aberration contributions from each surface, drives the design to an unideal freeform aberration balance. Whereas, the reflective prism is better able to balance the freeform aberrations from each surface reflection and thus leads to a better design for examining the effects of GRIN for color correction in the prism geometry.
The homogeneous reflective prism design has a material of PMMA with four unique surfaces, two reflective surfaces, and two refractive surfaces, as shown in Figure 2-1. PMMA is chosen as it is a common material for prism designs, but it is also one of the base materials for the GRIN material used in this study. This allows the design form not to change drastically when replacing the homogeneous material with a GRIN material. Fringe Zernike polynomials are used to describe each surface. There is not total internal reflection at any of the surfaces, rather the right surface would be partially coated to allow reflection and transmission, while the top surface would be coated for 100% reflection.
Figure 2-2 shows the ray aberration plots for the homogeneous reflective prism. Only a few of the 23 fields are shown, including the fields around the optical axis, as well as the edge of the field. The limiting aberration in the system is lateral color, which is expected since the system is composed of a single material, PMMA, and no symmetry around the pupil or image.
Figure 2-3 shows the MTF of the system with 23 fields. The performance does not meet the desired > 10% at 50 lp/mm. It does however achieve > 10% at 30 lp/mm, which is often the minimum specification required. The MTF chart does not give the full picture of the performance as there can be other fields which are worse than the ones plotted. Figure 2-4 shows the full field displays of the sagittal and tangential diffraction MTF at 50 lp/mm. The plots show only half of the y fields because the design is in XZ plane symmetry, so only half of the y field needs to be shown to get a full picture of the system. These plots give a whole picture of the MTF performance across the field and show that the homogeneous system is not quite meeting the specification at 50 lp/mm.
The GRIN reflective prism design utilizes a GRIN material composed of PMMA and polystyrene. This GRIN material has been characterized and manufactured extensively at the University of Rochester [4, 15]. It has a GRIN abbe number of 9 and maximum change in index of Δn ≈ 0.1. While the University of Rochester’s GRIN polymer is not developed for 3D printing, it is composed of similar polymers to those being used for 3D printing. The goal of this design it to see if a semi-arbitrary GRIN material could improve the design. The difficulty in the design lies in implementing a GRIN geometry that is not the standard axial, radial, or spherical GRIN, but rather some combination of all three. The other challenge is ensuring that the design software is tracing through the material properly, as there are multiple reflections through the same volume. Once these issues have been solved, the design is mostly a matter of exploring possible GRIN geometries as well as different material combinations. Figure 3-1 shows the final design of the GRIN reflective prism that looks almost identical to the homogeneous design. The right figure shows the GRIN profile overlaid on the drawing of the lens with a color bar to show the index variation, which is the major difference in the design. The change in index is Δn ≈ 0.013 which is a relatively small change in index compared to the total possible.
With the two lens drawings side-by-side, an idea as to how GRIN can improve the design becomes evident. The first pass through the GRIN adds power as it is acting like a spherical GRIN. After the first reflection, the rays and the index change are almost parallel, so the GRIN is not doing much at all. But after the second reflection, the GRIN is acting like a radial and axial combination. So with this prism and GRIN geometry combined, the system effectively has two different GRIN profiles in one. The freedom of the GRIN allows the system to tailor when the rays see index change, in this case through the first and last pass through. Figure 3-2 shows a subset of all the ray aberration plots, and instead focuses on the most important on-axis and full-field plots. From these plots, the lateral color in the GRIN design is reduced significantly compared to the homogeneous design. The system is now limited by secondary color rather than lateral color.
Figure 3-3 shows the MTF of the GRIN design. All of the plotted fields are meeting the > 10% at 50 lp/mm, which is the targeted specification. There are 25 fields now as there were a couple points in the full field MTF that were not above the 10% requirement. Again, the MTF chart does not give the full picture of the performance as there can be other fields which are worse than the ones plotted
Figure 3-4 shows the full field displays of the sagittal and tangential diffraction MTF at 50 lp/mm. The plots show only half of the y fields because the design is in XZ plane symmetry, so only half of the y field needs to be shown to get a full picture of the system. These plots give a whole picture of the MTF performance across the field and show that the GRIN system is not quite meeting the specification at 50 lp/mm for one or two points. If the sagittal and tangential performances are averaged across the full field, the design would meet the specification. This performance is also evaluated at the full 8 mm eyebox, which would in practice not be used all at once. A more conventional assessment (not shown) would be to evaluate the full field displays over sub 3 mm effective pupils on center and decentered at multiple locations with in the 8 mm eye box. Our analysis over the full 8 mm eyebox is to emphasize the correction of this design and even point to potentially being able to extend the field of view given that the system is over performing in use.
A prism design with freeform GRIN was designed with a FOV of greater than 45°, eye relief of 18.25 mm, eyebox of 8 mm, and performance greater than 10% at 50 lp/mm. The design utilized the reflective prism rather than the TIR prism as the reflective prism ray paths made implementing a complex GRIN profile simpler. The GRIN is able to reduce the amount of lateral color in the system which was previously limiting the homogenous design.
While the University of Rochester’s PMMA/polystyrene GRIN material in its current manufacturing process is not developed for 3D printing, the advances in additive manufacturing for polymers, a similar polymer could be developed. This work has shown that GRIN has significant potential for improving these prism design forms with its ability to bend rays within the material. Plans for continuing to explore the potential of GRIN in freeform prisms include: exploring materials which are 3D printable, increasing the field of view while sampling 3 mm pupils over the 8 mm eyebox, implementing GRIN into the TIR prism geometry, and using GRIN in monochromatic systems for adding power from the material for reduced volume. While this study looked at the potential of GRIN for color correction, especially lateral color, it has also shown that GRIN in freeform prism designs has a larger potential for improving key factors in improving the performance and usability of augmented and mixed reality systems.
The authors would like to thank Synopsys for the student licenses to Code V. The primary author would also like to acknowledge the National Science Foundation Graduate Research Fellowship (Grant No. DGE-1419118) for their support.
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