We propose a see-through display with a wavefront coding (WFC) technique using a cubic phase plate to extend the depth of field (DOF) of the projected virtual image. The image projected by the WFC see-through display allows the eyes to see an object in the range of accommodation and a clear projected image simultaneously without re-focusing. The DOF of the clear projected image by the system can be extended from 250 mm to an optical infinity of 10 m.
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See-through displays offer digital images projected onto alongside forwarding field of view, which eliminates the inconvenience that occurs when the eyes refocus between the scene and device. With this advantage, wearable smart glasses, such as Google glass , are now being considered for applications in next generation optics . Smart glasses with touch gesture operations , high-speed cellular networks, and software applications may replace mobile phones in the future. Such wearable see-through displays are convenient to the user who can simultaneously see an image on the display with private information, such as the logins of a bank account or a personal social website, as well as the scene ahead, thereby increasing safety.
However, the constraint of the depth of field (DOF) for the projected image of a see-through display is an important issue that cannot be ignored. A see-through display with a fixed-focus image causes an eye reflex between the projected image and the scene, which decreases safety and efficiency and takes few milliseconds during the process. Earlier research  provided a method for projecting an image to extend the DOF by a fast focal sweep projection with a tunable lens, while another research  proposed smart eyeglasses with tunable eyepieces for autofocusing. However, such technologies for adjusting the optical power of a see-through display in smart glasses may suffer from extra power consumption for the battery operation hours and reduce the system speed.
The extension of the DOF is difficult to achieve using a traditional optical system because of the unknown amount of defocus [6–7]. As the defocus becomes increasingly severe, the point spread function size will increase, which causes a region of zero values to appear in the optical transfer function (OTF) and modulation transfer function (MTF). This results in a loss of spatial frequency information for the image. Fortunately, Dowski et al. proposed a wavefront coding (WFC) technique to extend the DOF for an incoherent imaging system . The wavefront can be modulated by a system with a cubic phase mask (CPM), cubic phase plate (CPP), and various CPM designs to form a uniform bundle of rays [6,9–15]. The WFC technique enables the OTF without zero values below the cut-off spatial frequency, which allows the intermediate image generated by a WFC system to be correctly restored by applying a digital filter [8,16], thereby obtaining a clear image. The WFC technique also allows the DOF to be extended up to 30 times that of the traditional criterion .
In this study, we developed a WFC see-through focus-free display with an extension of the DOF by the inverted WFC technique process shown in Fig. 1. The intermediate image is coded and obtained by applying a simple digital filter to the original input image in digital imaging processing. The process shows that the system projects an intermediate image and forms a nearly uniform wavefront distribution in a specific region. By using a concave mirror with phase modulation, the intermediate image is decoded using the system. The WFC see-through display combined with the clear projected image and extension of the DOF enables it to be seen while the eyes are focusing on any object in the range of accommodation.
2. Design of the WFC see-through display
The optical system created using the optical design software CodeV is shown in Fig. 2. The system contains the eye model and the WFC see-through display with the object plane, reflector, concave mirror, and beam splitter. The object plane is simulated as a 16 × 16 mm2 microdisplay with a pixel size of 10 μm and allows a resolution of 1600 × 1600 pixels so that the WFC see-through display can project a virtual image size of 306 × 306 mm2 with 23.6° in the horizontal and vertical directions at 700 mm in front of the eyes. The F-number of the system is 4.8, and the entrance pupil is 10 mm, which allows eye movements and fitting to different users to see the virtual image.
Considering the position of a CPP, we employ a third-order coefficient of the xy-polynomial surface (cubic coefficient) on the concave mirror with the radius of curvature of 161.63 mm, it avoids the issue that obscures and distorts the view of the real world if the CPP is placed in the view path, and it also eliminates one optical element. Therefore, the system generates a two-dimensional phase function given by Eq. (1),
Considering the situation where the larger the cubic coefficient, the more the rays deviate away, while the smaller the cubic coefficient, the more sensitive the system is to defocus . Different cubic coefficients assigned on the concave mirror provide different MTFs, the RMS values of the MTFs are shown in Table 1, the RMS values increase as the cubic coefficient increased. The MTF plots for C=0.0004, 0.0008, and 0.0012 are shown in Fig. 3, the cut-off frequency is corresponding to the MTF value of 21 cycles/mm. Thus, the cubic coefficient assigned for the phase modulation used in the design is taken as C=0.0008 to prevent the sensitive MTF and suppressive MTF. The marginal ray height is 1.67 mm (The equivalent coding strength for the cubic coefficient assigned in the pupil of the eye is C=0.00085 with the marginal ray height of 2.97 mm).
The MTF of the WFC see-through display shown in Fig. 3 has no zero values below the cut-off spatial frequency, which allows the intermediate image to be restored without a loss of spatial frequency information. The intermediate image with a nearly uniform wavefront distribution over a certain range is completely projected and decoded by the WFC see-through display, the DOF of the projected image can be extended, and a clear projected image can be seen even if the eyes are focused on nearby or distant objects. Although, the MTF shown in Fig. 3 is not as expected for traditional optical systems, the system with the MTF is used to decoded the intermediate image. The intermediate image for projection is obtained by using a simple digital filter to the original input image, and the foundation for digital image processing in the spatial frequency domain can be written symbolically as
3. Simulation Results
The simulation was performed to illustrate that the eyes can simultaneously see both the object at the range of accommodation and the image projected, clearly without re-focusing. The original input image for the simulation is shown in Fig. 5(a). An intermediate image was obtained after applying the digital filter, as shown in Fig. 5(b). A comparison of the see-through display with and without the WFC is shown in Fig. 6. The projected images with and without the WFC as seen by the eye are shown in Fig. 6(a) and 6(b) while focusing on a sidewalk sign at 700 mm. As Fig. 6(c) shows, when the eyes are focused on a distant sign at 10 m, they can clearly see the sign at 10 m as well as the projected image with WFC; however, the projected image without WFC, shown in Fig. 6(d), is blurred, which means that the eyes must refocus to 700 mm, the position of the projected image, by trading off the accommodation reflex time. When the eyes are focused on the nearer object at 250 mm, the projected image with WFC, shown in Fig. 6(e), is clearer than the projected image without WFC, as shown in Fig. 6(f). The simulation results show that the projected images with higher peak signal-to-noise ratio (PSNR) values are clear and sharp. The PSNR is defined as
To illustrate that the WFC see-through display can project near-identical images, we use the structural similarity index measurement (SSIM) to compare the projected and the original input image, while the eyes focus on objects at different distances, as shown in Fig. 7. The results show that the projected image with WFC at different distances has a near-identical SSIM value, and it shows high similarity. For the see-through display without WFC, the projected image has the highest SSIM value only when the eye sees it at a projection position of 700 mm.
In this study, we presented a WFC see-through display that can be applied to smart glasses. The simulation results show that the DOF of the projected image of the WFC see-through display was extended, allowing the projected image to be seen clearly when the eyes focused on objects at any distance. The projected images with WFC are clear and have high similarity, and its PSNR values are high, while the PSNR values for the projected images without WFC decreased. In addition, the see-through display without any autofocusing technique increases the efficiency and safety because the eyes do not spend time on the process of accommodation reflex.
The authors declare no conflicts of interest.
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon request.
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