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Abstract
We propose a large viewing angle integral imaging 3D display system based on a symmetrical compound lens array (SCLA). The display system comprises a high-resolution 2D display panel, an SCLA, and a light shaping diffuser. The high-resolution 2D display panel presents an elemental image array, the SCLA modulates the light rays emitted from the 2D display panel to form 3D images in space, and the light shaping diffuser eliminates the gaps between 3D pixels of the 3D images. We find that the lateral aberration is a crucial factor that affects the resolution of the reconstructed 3D image. The symmetrical structure of the SCLA enables a reduced focal length and the elimination of lateral aberration, improving the viewing angle and the 3D image resolution simultaneously. The experimental results confirm that the proposed display system increases the viewing angle to 68.6°, achieving a comparable resolution of the full field of view while maintaining a simple structure.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
With the advancement of science and technology, there has been an increase in demand for display technology. One of the most prominent areas of interest is the 3D displays that show depth information. Integral imaging (InIm) 3D display offers a number of advantages, including full parallax, quasi-continuous viewpoints, full-color capabilities [1,2], and without requiring specialized equipment [3,4]. Consequently, it is regarded as one of the most promising 3D display technologies [5–7]. The concept of InIm has been proposed by Lippmann in 1908. It is composed of two processes: 3D data acquisition and 3D image reconstruction [8,9]. In the 3D data acquisition process, the light field information is captured by a micro-lens array (MLA) and a recording medium, resulting in an elemental image array (EIA) that contains both the spatial and the angular information. Then, during the 3D image reconstruction process, the EIA is displayed on the 2D display, and the MLA is precisely aligned with the EIA to reconstruct 3D images. InIm enables viewers to perceive 3D images with depth from various viewing positions. However, InIm still suffers from the issue of limited viewing angles [10–12], which leads to image artifacts and crosstalk for viewers outside the viewing angle region, significantly reducing the overall viewing experience.
In the evaluation of the visual performance of the InIm 3D display, the viewing angle is often a key indicator. To expand the viewing angle of the InIm 3D display, optical design techniques [13,14] are commonly employed to optimize the MLA. A curved MLA has been employed to extend the viewing angle, as it makes the off-axis elemental image at the center of the elemental lens [15], as well as the on-axis elemental image. However, at larger viewing angles, there is severe crosstalk in the reconstructed 3D image because the MLA is not able to accurately couple to the EIA at the edge positions. A large-scale InIm 3D display system employing an aspheric MLA has been proposed [16]. The aspheric MLA is used to modulate the vertical light field information in the horizontal direction, thereby improving the horizontal resolution and increasing the number of viewpoints. This method achieves a horizontal viewing angle of 40° at the expense of the vertical 3D effect. To increase the utilization of light rays emitted from the 2D display panel for 3D reconstruction, an InIm 3D display system based on an MLA coupled with a dual-prism array has been presented [17]. The dual-prism array is coupled with the MLA to capture light rays from a wider angle, thereby increasing the refracting capabilities of the lens array. Compared to the conventional InIm without the dual-prism array, this system achieves a threefold enhancement in the viewing angle. However, this system uses two layers of optical modulation devices, resulting in decreased image resolution and a lower brightness in the reconstructed 3D image.
There is an upper-performance limit to optimizing the MLA, and the total information displayed in the 2D display limits the number of available lights for reconstruction [18,19]. To overcome the constraints imposed by the limited information capacity of the 2D display, researchers utilize space-division multiplexing or time-division multiplexing schemes to enhance the information capacity and therefore improve the viewing angle [20–23]. For example, an InIm 3D display system utilizing two overlapped panels has been proposed to enhance the viewing angle [24]. This system integrates the viewing areas of both panels to enhance the overall system information by organizing effective and transparent information on both panels. However, the attenuation of the light causes a reduction in the brightness of the reconstructed 3D image. Collimated backlight sources can precisely control the propagation of light and limit the rays to a narrower divergence angle to pass through multiple specific lenses. This process enables precise 3D image splicing in space-time multiplexing. Consequently, an InIm display system based on a collimated backlight has been reported to enhance the viewing angle of 3D displays [25]. The system includes three sets of collimated backlights, an LCD screen, and an MLA. Three sets of collimated backlights are used to control the direction of light propagation and are time-division multiplexed in conjunction with a high-refresh-rate 2D display to realize the splicing of viewing areas. The system provides a 120° viewing angle. Tracking the precise location of the viewer can provide more accurate data, thereby improving the viewing angle performance of the InIm display [26]. The EIAs are dynamically switched according to the viewer's head position, resulting in a wide viewing angle. However, these systems using both space-division and time-division multiplexing mentioned above would introduce additional optical and circuit equipment.
We propose a large viewing angle InIm 3D display system based on a symmetrical compound lens array (SCLA). The SCLA consists of multiple compound lens units, each comprising two symmetrically structured lenses. This results in an enhanced viewing angle of 68.6°. The SCLA possesses a large aperture and a short focal length, which reduces the object distance of the lens array and enhances the viewing angle. The symmetrical structure eliminates lateral aberrations in the lens array and improves the resolution of 3D images at wider viewing angles. The SCLA has the advantages of a simple structure, cost-effectiveness, and feasibility.
2. Principle
2.1 Schematic diagram of the proposed InIm display
The schematic diagram of the conventional InIm display is depicted in Fig. 1(a). The conventional InIm display consists of a 2D display panel, a single lens array, and a light shaping diffuser. The 2D display panel is used to display the EIA. The emitted light is refracted by the MLA to converge near the central depth plane, creating a discrete 3D image. The position of the central depth plane is derived from the formula of Gaussian imaging. The light shaping diffuser is located at the central depth plane to eliminate the lenslet grid which is caused by the non-illuminated areas between the lens array. The diffusion angle of the light shaping diffuser is determined by the pitch of the lens array and the distance between lens array and the light shaping diffuser [27]. The viewing angle of the InIm display defines the angle at which a smooth and continuous 3D image can be observed without any visual distortion. The pitch of the MLA and the object distance from the 2D display panel to the MLA, as shown in Fig. 1(b), limit the viewing angle. According to the geometric relations, the viewing angle θ is expressed as
where P is the pitch of the MLA, and g denotes the object distance of the MLA, which is the distance from the object principal plane of the MLA to the 2D display panel. The single lens is considered a thin lens, with the object principal plane coinciding with the image principal plane, and both planes are positioned at the thickness center of the lens. For the reconstructed 3D image to be an enlarged real image, g must meet the following conditions where f is the focal length of the single lens. It can be seen from Eq. (1) that increasing the pitch P and reducing the object distance g can increase the viewing angle θ.
Fig. 1. Conventional InIm display. (a) Schematic diagram of the conventional InIm display based on the single lens array. (b) Principle of 3D image reconstruction.
The schematic diagram of the proposed InIm display system with an improved viewing angle is illustrated in Fig. 2(a). It includes a 2D display panel, an SCLA, and a light shaping diffuser. The SCLA consists of several large-aperture, short-focal-length symmetrical compound lens (SCL) units. The SCL units are arranged periodically. In each SCL unit, two lenses of the same focal length f are positioned equidistant from the aperture stop. The diaphragm is positioned at the center of the two lenses, and the front and rear lenses are symmetrical in surface shape about the diaphragm. The focal length f’ of the SCL unit is given by
where d is the distance between the optical centers of the two symmetrically arranged lenses, and d nearly approaches zero, resulting in f’ being approximately half of f. If the distance d falls between zero and f, f’ is shorter than f. Compared to the single lens, the SCL has a shorter focal length but increased thickness. Consequently, the object principal plane is not aligned with the image principal plane. The object principal plane shifts from the center of the compound lens towards the 2D display panel, as shown in Fig. 2(b). The object distance of the SCL units is g’. It is evident from Eq. (2) that g’ needs to satisfy the following formula ofFig. 2. Proposed InIm display system. (a) Schematic diagram of the proposed InIm display based on the SCLA. (b) Principle of 3D object reconstruction with an improved viewing angle θ’.
Since f’ is less than f, g’ must be less than f. Therefore, the viewing angle θ’ is expressed as
From Eq. (2), it is evident that decreasing g’ causes θ’ to increase, indicating that a large pitch can be achieved while reducing the object distance by utilizing an SCLA, ultimately enhancing the viewing angle.
2.2 Impact of MLA aberration on 3D images
The pixels of the 2D display panel are merged via the MLA to produce a 3D image, as shown in Fig. 3(a). The lens aberration of the MLA could significantly reduce the resolution of the 3D images, especially in the large viewing angle InIm display system..
Fig. 3. Voxel reproduction of the InIm display. (a) Schematic diagram of the voxel formation in 3D images. (b) Spot size of different pixels on the image plane.
Lens aberrations are divided into longitudinal and lateral aberrations. Longitudinal aberration is caused by the change in refractive index for light at different aperture angles and its magnitude is related to the lens aperture. Lateral aberration is caused by the asymmetrical imaging of an off-axis object by the optical system, and its magnitude is related to both the field of view of the object and the aperture of the lens. The voxel reproduction within the InIm display process is illustrated in Fig. 3(b). Light rays emitted from three pixels X, Y, and Z pass through the corresponding lens units A, B, and C, respectively, and converge to form a voxel on the imaging plane. The longitudinal aberrations of the three pixels imaged by corresponding lenses are equal because they have the same surface shapes. The fields of view of pixel X to lens A, pixel Y to lens B, and pixel Z to lens C are ωx, ωy, and ωz, respectively. And their lateral aberrations are denoted as δTx, δTy, δTz, respectively. The absolute value of the field of view for pixels X and Z is equal, which is denoted by $|{{\omega_x}} |= |{{\omega_z}} |$. The lateral aberrations can be expressed as
It is evident that the pixels of the 2D display have different lateral aberrations as they pass through different lens units. The lateral aberrations become more severe as the field of view expands, leading to enlarged imaging spots. The enlarged imaging spots converged to form a large voxel, which rapidly degrades the resolution of 3D images. The resolution of InIm is primarily limited by lateral aberrations. It should be noted that the pixel imaging spot in the large field of view is considerably larger than that in the central field of view. To enhance the resolution of 3D images, it is essential to correct the lateral aberration of the MLA, especially the lateral aberration in the large field of view.
2.3 Aberration analysis of the SCL
To enhance the viewing angle of the InIm display and maintain a high-resolution 3D image, it is essential to correct the lateral aberration. Conventional techniques for lens optimization include the assembly of multiple lenses to form a compound lens unit or the replacement of spherical lenses with aspherical lenses. However, the design of multiple lenses increases the thickness of the MLA or the complexity of the system. And in the process of optimizing the lens, it is not always possible to shorten the focal length and increase the aperture since it causes the lens aberration to rise. Using aspherical lenses instead of spherical lenses poses challenges related to complex and costly processing. Neither of the above design methods is suitable for improving lateral aberration individually. Therefore, we design a new symmetrical structure lens to achieve a short focal length without increasing the aberration and remain the aperture unchanged. The symmetrical structure of the optical system can correct the lateral aberration, thereby improving image quality at wide viewing angles. According to the aberration theory, lateral aberration consists of coma and distortion, and the main component in aberration is primary aberration. The primary coma KT ‘ in the meridional plane for the compound lens is mathematically expressed as
The imaging optical path for calculating the primary meridian coma aberration of the SCL is illustrated in Fig. 4(a). The SCL comprises two lenses - lens1 and lens2, which have identical parameters, and a diaphragm. As lens1 and lens2 are symmetrically positioned relative to the stop, the primary aberration distribution coefficients on the refractive surfaces are equal. The aberration of the whole SCL is the combination of the aberrations of lens1 and lens2. For ease of calculation, we calculate the primary meridian coma aberration of lens1 using the forward light path, and the primary meridian coma aberration of lens2 using the reverse light path, as depicted in Figs. 4(b) and 4(c). The primary meridian coma K'1 calculated in the forward direction for lens1 is expressed as
and the primary meridian coma K'2 computed in the reverse direction for lens2 is expressed asFig. 4. Optical path diagram for the primary meridian coma aberration calculation. (a) Optical path diagram of the SCL. Optical path diagram of (b) forward aberration calculation for lens1 and (c) reverse aberration calculation for lens2.
The primary meridian coma K'c of the entire SCL is expressed as the difference between lens1 and lens2, which can be calculated as
To produce an enlarged real image, the image distance of the lens unit in the InIm 3D display must exceed the object distance. Figure 4(a) shows that the object distance for lens1 is l, and for lens2 in the reverse calculation is l’. Since lens1 and lens2 have the same focal length, it is obvious that the image distance for lens1 exceeds that of lens2. The difference ΔK in primary meridian coma aberration between the SCL and the single lens1 is calculated as
Since ${u^{\prime}_r} > 0$, the variance ΔK < 0, which means the primary meridian coma aberration of the SCL is smaller than that of the single lens. The above results show a reduction in coma aberration of the SCL.
The use of a symmetrical structure in the design of the compound lens corrects the lateral aberration at the expense of introducing longitudinal aberration. Primary spherical aberration is the most important of the longitudinal aberrations. To illustration, the primary spherical aberration δL’ is expressed as
In an SCL, the primary spherical aberration δL'c is expressed as the sum of the primary spherical aberrations of lens1 and lens2, which is calculated as
The difference ΔL in primary spherical aberration between the entire primary SCL and the single lens1 can be expressed as
The difference ΔL is positive, which means that the primary spherical aberration of the SCL is larger than that of the single lens. Primary spherical aberration can be minimized by reducing the object aperture angle. The relationship between the primary spherical aberration of the lens and the aperture angle can be mathematically expressed as
where a1 denotes the primary spherical aberration coefficient, a2 denotes the secondary spherical aberration coefficient, and u1 indicates the object aperture angle. Equation (18) shows that the primary spherical aberration is directly proportional to the square of the object aperture angle. Therefore, the primary spherical aberration of the SCL can be reduced by reducing the aperture angle.An SCL is designed, as shown in Fig. 5(a), composed of two spherical plano-convex lenses and a small aperture stop. A comparative analysis has been made of the primary coma distribution coefficient of three types of lenses: the conventional single lens, the conventional three-piece compound lens [28], and the SCL. These three types of lenses have the same aperture. The results of the primary coma distribution coefficients at different angles are shown in Fig. 5(b). The proposed SCL has the lowest primary coma distribution coefficients which provides evidence that the lateral aberration of the SCL is significantly optimized within the field of view. The spot diagrams of three types of lenses within a viewing angle of 0° to +34.3° are shown in Fig. 5(c). The root mean square (RMS) spot size is related to several factors so it is not possible to directly compare the aberration coefficients of the two kinds of lens based on RMS spot size. Here we use the spot diagram to compare the lens’ light-gathering ability. The RMS spot size of SCL is the smallest, which means that the symmetrical structure has a greater ability to gather light and a smaller distribution of spots.
Fig. 5. (a) Front view of the SCL. (b) Comparison results of the primary coma aberration. (c) Spot diagrams of the SCLs.
3. Experimental results
We developed a prototype of the InIm display system to demonstrate the enhanced viewing angle and resolution of 3D images, as shown in Fig. 6. The prototype consists of a 5.5-inch LCD panel, an SCLA, and a light shaping diffuser. The LCD panel has a resolution of 3840 × 2160 pixels with each pixel measuring 0.031 mm. The lens array comprises 5 × 10 lens units evenly distributed horizontally and vertically with a pitch of 10.8 mm. Each lens unit comprises two identical plano-convex lenses with a diameter of 10 mm and a focal length of 15 mm, with the convex surfaces facing each other. The distance d between two plano-convex lenses is negligible, resulting in a combined lens unit with a focal length of 8 mm. The diffusion angle of the light shaping diffuser is 20°, which is determined by the pitch of the lens array and the distance between the lens array and the light shaping diffuser. The comprehensive parameters of the prototype are presented in Table 1.
Table 1. Configuration of the InIm display system prototype
Figure 7 shows the reconstructed 3D images observed from different angles by using the prototype. The 3D scene comprises two cube models arranged in space. An off-axis camera array is used to capture the 3D scene and then the light field information is recorded in the 3D data acquisition process. The EIA is generated and loaded onto the LCD panel after the light field information is processed using the encoding algorithm. The 3D images maintain seamlessness in both horizontal and vertical directions, ranging from -34.3° to +34.3°, as shown in Figs. 7(a) and 7(b), respectively. Figure 8 shows the reconstructed 3D images of the InIm display system based on the conventional single lens array. The viewing angle of it is narrower than that of the proposed InIm display system.
Fig. 7. 3D images of the prototype observed from different angles in (a) horizontal direction (see Visualization 1) and (b) vertical direction (see Visualization 2).
Fig. 8. 3D images of the InIm display based on the single lens array observed from different angles in (a) horizontal and (b) vertical directions.
However, the motion parallax information of the reconstructed 3D images is week. The root cause of it is that the symmetrical compound lens array possesses a small depth of field due to a limited amount of whole displayed information. It is not possible to increase all performance parameters simultaneously. The blurring effect of the light shaping diffuser may also lead to a loss of parallax information because it may cause adjacent views to merge into one view.
A comparison result of the 3D imaging qualities between the InIm 3D display systems utilizing the single lens array and the proposed SCLA at different viewing angles of 0°,±15°, and ±25° is shown in Fig. 9. A 1951 United States Air Force (USAF) resolution test chart is employed to test the imaging qualities. The experimental results show that the resolution of the proposed InIm display depicted in Fig. 9(a) is greater than that of the InIm display using a single lens array, which is illustrated in Fig. 9(b). In addition, the resolution of the proposed InIm 3D display system decreases at a slow rate as the viewing angle increases. This confirms that SCLA has minimal lateral aberration. The single lens has a large lateral aberration, so its resolution decreases seriously as the viewing angle increases. The proposed SCLA can effectively reduce lateral aberration and improve the 3D image resolution.
4. Conclusion
This paper proposes a large viewing angle InIm 3D display system with a wide viewing angle based on a symmetrical compound lens array. The lens has a symmetrical design with a large size and a short focal length. This design ensures a large viewing angle while preventing increase in aberration and even reducing the lateral aberration, therefore enhancing the resolution of the reconstructed 3D images with a large viewing angle. The proposed display system enables 3D images to be reconstructed with an extended viewing angle of 68.6°. It has a compact and lightweight design compared to the conventional InIm display with a compound lens array. This makes it suitable for applications of portable and downsized 3D display devices. However, the depth of the reconstructed 3D image is small, causing the parallax relationship of the reconstructed 3D image to be inconsistent with the actual situation, and the 3D effect is not obvious. Future work will focus on improving the depth of the 3D image and enhancing the three-dimensional sense of the image.
Funding
National Key Research and Development Program of China (2022YFB3608200); National Natural Science Foundation of China (62105014, 62105016).
Acknowledgment
The model used in Figs. 7 and 8 and in Visualization 1 and Visualization 2 is modified from Simple Rubix Cube by Blender3D, licensed under a Creative Common Attribution 4.0 License.
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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