Compact zooming optical systems for panoramic and telescopic applications based on curved image sensor

Abstract. Biological eyes in nature have strongly inspired novel optical systems. In this regard, imaging systems mimicking fish eyes and human eyes have been reported for having a wide field-of-view (FoV) and relatively higher magnification properties, respectively. However, most of these systems have complex lens configurations because of their flat image sensors. As these optical systems are bulky, heavy, and expensive, they have limited application in small devices (e.g., drones and mobile phones). In addition to a simplistic design, multi-functionality is essential for broad applications. Therefore, this study proposes a compact zooming optical system (CZOS) that combines the properties of natural vision systems (i.e., human and fish eyes) using a curved focal plane. The CZOS controls the zoom range through the modification of the distances between the single front lens (i.e., negative meniscus) and dual rear lens (i.e., bi-convex/positive meniscus lenses) groups. In the proposed system, the panoramic mode had an FoV of 200 deg and a magnification of 0.23, whereas the conversion of the system to the high magnification mode increased the magnification over two times with an FoV of 70 deg. These promising results demonstrate that the proposed simple imaging system is applicable to small-scale electronics.


Introduction
Biological imaging systems of animals have developed through evolution. To promote the survival of organisms in their habitat, the imaging systems of creatures are differently optimized, e.g., for viewing angles and depth perceptions. [1][2][3][4][5] Consequently, because of their unique system and functionality, natural eyes have inspired the development of biomimetic cameras. [6][7][8][9][10][11][12][13][14] Regarding bio-inspired systems, those mimicking fish eyes have been widely studied to achieve the characteristics of a wide field-of-view (an FoV of ∼180 deg) with high visual acuity. [15][16][17] In addition, systems based on an imitation of human eyes, which have relatively narrower viewing angles (an FoV of ∼60 deg) than fish eyes, have been researched for the advantages of low optical aberration and high resolution. [18][19][20] As most conventional imaging devices employ flat image sensors, they require a multitude of lenses (7)(8)(9)(10)(11)(12) to obtain high-quality visual information, thereby resulting in complex and expensive optics. [21][22][23] For instance, a prototype fish eye system, offering a wide FoV (∼180 deg), comprised of multi-lens optics (>6 lenses) and a flat image sensor to minimize optical aberration. 24 In addition, a reported biomimetic model of the human eye based on a flat sensor required several lenses to remove the field curvature and distortion. 25 Meanwhile, curved image sensors have been developed to address several drawbacks of flat image sensors by deforming flexible and stretchable planes into curved forms. [26][27][28][29][30] Because the Petzval surface is a curved shape, the use of a curved image sensor facilitates a high image quality with a smaller number of lenses than a flat sensor. 31 Thus, an optical system with a curved image sensor can be closer to natural vision systems with a compact design and higher resolution than traditional imaging devices. Although recent research has yielded a complexity reduction of bio-inspired devices through the use of curved sensors, 17,20,32,33 several designs provide a limited function (i.e., fixed zoom range) because of the single-lens system. For advanced applications, such as drones and automobiles, multi-functionality of an imaging system is as essential as compactness. However, studies on bio-mimetic systems considering both compactness and functional versatility are rare.
This study proposes a compact zooming optical system (CZOS) that simultaneously reduces the complexity and provides continuous zooming modes based on curved image sensors. The CZOS can vary the zoom range continuously (70 deg to 200 deg) through the modification of the distances between the single front and dual rear lens groups, which mimics two vision systems (i.e., human and fish eyes). Further, to examine the applicability of the CZOS as the zoom lens, a simulation tool based on geometrical optics (Zemax) was used to conduct several simulations: ray-tracing, spot radius, relative illumination (RI), field curvature, angular magnification, and geometrical image analysis. The specific lens variables (e.g., thickness, radius curvature, and the distance between each lens) of the proposed optical system were optimized by utilizing constrained merit functions in the simulation tool. In addition, the red, green, and blue (RGB) light corresponding to the wavelengths of 450, 550, and 650 nm, respectively, were set as the fundamental wavelengths. Figure 1(a) shows a schematic demonstrating the anatomy of the teleost eye (top) and human eye (bottom). For a wide FoV (∼180 deg) with clear visual acuity in dim underwater conditions, the imaging systems of aquatic mammals include a single spherical monocentric lens and a highly light-sensitive hemispherical retina. 17 Compared with aquatic vision, the natural human eye comprises a small bi-convex lens with a small curvature, which results in a narrower FoV (∼60 deg) and higher magnification.  traditional panoramic and telescopic imaging systems based on flat image sensors. Although conventional designs have complex lens arrays, the functions of these systems are similar to those of natural eyes. The designs of both systems are provided in the Zemax software. The simulated results demonstrated the wide FoV of panoramic lenses and high magnification of telescopic lenses; however, both systems required highly complex lens configurations to flatten the field curvature onto the flat image sensor. Because large form factors of systems hinder their advanced application for devices such as drones and mobile phones, a more compact design should be fabricated by reducing the number of lenses.

Results and Discussion
The proposed CZOS, composed of three distance-adjustable lenses and a curvilinear image sensor, is shown in Fig. 2(a). Through the introduction of the curved image sensor, the complexity of an optical system can be significantly reduced compared with that of conventional biomimetic imaging systems. 29 Furthermore, the angular magnification and FoV can be varied by changing the distance between the front and rear groups (D 1 ). The lens configuration of the CZOS was optimized to achieve the dual-mode (i.e., panoramic and telescopic) using constrained merit functions in software. Although the variation of D 1 can provide multiple zoom modes (Fig. 5), these modes were simplified into dual modes for a simpler explanation. The specific lens variables such as thickness, material, semi-diameter, and radius of curvature (ROC) are described in Table 1. The shape of aperture is curved to avoid overlapping between lenses, as shown in Fig. 6. In the panoramic mode of the CZOS, the front and rear lenses were placed at the proper distance (D 1 : 41.2 mm) to achieve a wide FoV similar to a traditional aquatic imaging system [ Fig. 2(b)]. Further, compared with the conventional design, as shown in Fig. 1(b), the panoramic mode of the CZOS also offers a wide viewing angle (∼200 deg) even for a small form factor (three lenses) because of the curved image sensor. When the distance between the front and rear lenses (D 1 ) was decreased to 5 mm, the FoV and magnification became narrower and higher, respectively, which is the telescopic mode of the CZOS [ Fig. 2(c)]. As the distance variation between the front and rear groups altered the size of a light beam, the FoV and angular magnification varied with D 1 . Consequently, the FoV of the telescopic mode decreased to 70 deg because of the smaller beam size. Thus, these results demonstrate that the variation of zoom range works well in a few lenses because of the curved image sensor. To optimize the ROC of the image sensor (R ima ) in the proposed system, the effect of the shape of sensors in the system was examined [Figs. 2(d) and 2(e)]. Because the illumination and spot size are considered to be essential factors in optical systems, RI and root-mean-square (RMS) spot radius were calculated as a function of the incident angle at a wavelength of 550 nm. The evaluation of RI and RMS spot size at different wavelengths (i.e., 450 and 650 nm) are shown in Fig. 7. Because the low RI and large spot radius in an imaging system hinder obtaining high-quality images, a high value of RI and small spot size should be achieved at a wide incident angle. As shown in Fig. 2(d) (left), the illumination increased with a higher curvature (i.e., small R ima ) of the image sensor in the panoramic mode (D 1 : 41.2 mm). Further, as the image sensor of the higher curvature approached the incident light from an object at a higher viewing angle, the RI gradually increased with the higher curvature. Thus, the highest curvature (R ima : 10 mm) provided the best illumination property, whereas the flat image sensor exhibited the lowest RI. Moreover, the tendency of spot size variation differed from that of the RI as the spot radius decreased with decreasing ROC up to 50 mm and then increased up to 10 mm [ Fig. 2(d); right]. This result indicates that the smallest spot size in the case of ROC is 50 mm because the shape of the focal plane is closer to the ROC of 50 mm. The ROC of 50 mm exhibited a relatively uniform and small spot radius than other shapes for wide angles. Therefore, 50 mm is the optimal ROC for achieving high-resolution imaging in the panoramic mode. In addition, the RI and spot radius size were analyzed in the telescopic mode, with the distance between the front and rear groups (D 1 ) being 5 mm [ Fig. 2(e); left]. The illumination moderately increased with a higher curvature, similar to the panoramic mode, that is, the higher curvature sensor approached incident light at a wide angle regardless of D 1 . However, the spot radius variation in the telescopic mode slightly differed from that in the panoramic mode [ Fig. 2(e); right]. The results showed a similar tendency with all curvatures except for the highest (R ima : 10 mm). When the distance between the image sensor and the object was reduced, the spot radius size at a large incident angle varied in a less sensitive manner with the curvature; however, an ROC of 10 mm was a sufficiently high curvature to change the spot size. As the telescopic mode of the CZOS with R ima : 50 mm exhibited better performance in RI and spot radius, this was selected as the optimal ROC. In addition, these results demonstrate that a curved image sensor exhibits better performance in optical systems than a flat image sensor. The optical performance can be further improved if using a curvature variable image sensor because the optimal curvature of image sensor varies across the zoom range (Fig. 8). Further optical analyses such as distortion the modulation transfer functions (MTFs) are shown in Figs. 9 and 10, which demonstrate that the optical performance can be improved using a curved sensor (R ima : 50 mm).
High resolution and uniform performance over a multi-spectral range are necessary for highquality imaging. In particular, red, green, and blue (RGB) wavelengths have been used for estimating performance in previous research as most digital images comprise pixels that are a combination of the RGB colors. 34,35 To analyze the applicability of the proposed design as an imaging device, the RMS spot radius of CZOS was calculated at RGB wavelengths (450, 550, and 650 nm), as shown in Fig. 3(a). D 1 was set to 41.2 mm, which is the panoramic mode of CZOS, to evaluate the performance at a broad angle. For a fair comparison, the CZOS using a flat image sensor was re-optimized (D 1 : 41.2 mm, D 2 : 7.1 mm, and D 3 : 4.01 mm) with the same angular magnification as the CZOS based on a curved sensor. Compared with the planar sensor, the spot radius of the curved image sensor (R ima : 50 mm) was similar at low incident angles (<50 deg); however, it was smaller at high incident angles (>50 deg). This result demonstrates that using the curved sensor resulted in a better and more uniform performance at a wide angle than the planar sensor even at different visible wavelengths (450, 550, and 650 nm). The spot radius diagram of the CZOS with different curvatures of the sensor is shown in Fig. 11. Further, the field curvature of CZOS at RGB wavelengths was also simulated with different shapes of the image sensors (i.e., flat and curved) [ Fig. 3(b)]. For the flat image surface, the result yielded positive values at a broad incident angle [ Fig. 3(b); top]. This result indicates that the focal plane was located behind the image plane at all incident angles. In contrast, the field curvature of the curved image sensor was closer to the black horizontal line at all wavelengths [ Fig. 3(b); bottom], thereby indicating that the optimized curved sensor was more suitable for imaging as the focal plane of the CZOS was closer to a curved design than a flat design. Figure 3(c) shows the simulated results of the point spread function (PSF) for different sensor shapes (flat and curved) to analyze the imaging resolution. The simulated PSFs of curved sensors exhibited smaller and uniform beam sizes at a wide angle. These results demonstrate that a curved sensor facilitates a better and more uniform performance of imaging at RGB wavelengths, implying a higher image quality of CZOS because the focal plane is a curved shape. In addition, through the introduction of aspherical lenses, the optical performance (e.g., spot radius, RI, field curvature, and MTF) can be further improved (Fig. 12). The aspheric coefficients of re-optimized CZOS are given in Table 2.
Observation of distant objects requires excellent resolution and high magnification in optical systems. The above results proved that the proposed CZOS possesses a reasonable optical quality for imaging, and the magnification performance should be evaluated. Thus, to analyze the zooming ability of the CZOS, the angular magnification was calculated for varying D 1 values [ Fig. 3(d)]. With decrease in the distance between the front and rear groups, the angular magnification increased because of the change in the focal length. Further, the angular magnification of the telescopic mode reached a value twice as high (0.47) as that of the panoramic mode (0.23). This result confirms the fine magnification function of the CZOS through the modification of the distance between two lens groups. Figure 3(e) shows the simulation results of the geometric image at different D 1 values (41.2 and 5 mm). A geometric image analysis was performed to confirm both the chromatic aberration and zooming properties of the CZOS. The points of the corresponding color represent the geometric images of each wavelength. The simulated image in the telescopic mode (D 1 : 5 mm) yielded a bigger letter than that in the panoramic mode (D 1 : 41.2 mm), thus demonstrating the magnification capability of the CZOS. In addition, the points of each wavelength were uniformly arranged as the CZOS exhibited good performance in the spot radius and field curvature analysis. Thus, these results demonstrate the zooming function of the CZOS and low chromatic aberration in both panoramic and telescopic modes. Figure 4 shows the results of an image simulation for an object with varying distances between the front and rear groups. In this process, the sizes of the systems were adjusted to obtain an effective focal length (EFL) of 6.2 mm, and the F-number of 5 was used. Further, the object height was set by D 1 to correspond to each FoV. Compared with the original image [ Fig. 4(a)], the simulated image in the panoramic mode (D 1 : 41.2 mm) exhibited a wide FoV, which almost fully yielded the original image [ Fig. 4(b); left]. The middle of Fig. 4(b) shows the magnified image at 20 mm of D 1 , which shows a narrower FoV than the panoramic mode. However, a much-magnified image was obtained when the front and rear groups approached 5 mm [ Fig. 4(b); right]. These results practically demonstrate the simple and continuous magnification method of the CZOS and high image quality of a curved focal plane. In addition, the lenses of the CZOS can be replaced with a plastic material, as the performance of the plasticbased system (p-CZOS) is similar to the CZOS (Fig. 13, Table 3). These results demonstrate the universality of our system.

Conclusions
This study investigated the applicability of the CZOS with curved image sensors as a continuous zooming system through ray-tracing simulations. Optical analysis factors, such as the RI and spot radius size, provided a design rule for the optimum ROC of the image sensor (i.e., 50 mm). The calculated results indicated that the curved image sensor reduced the complexity of the CZOS while maintaining low optical aberration. In addition, the simulation results of field curvature and spot radius at different wavelengths (450, 550, and 650 nm) demonstrated that the CZOS yielded a low chromatic aberration and consistent optical performance at a wide viewing angle. Finally, the image analysis demonstrated the high imaging performance of the CZOS in the variation of the FoV (70 deg to 200 deg). These results indicate that the proposed imaging system exhibits a reasonable optical quality for imaging and magnification function. The simple imaging system is expected to be applicable to compact devices such as drones and automobiles.

Appendix
We expect that the data in the appendices will benefit the reader's understanding of the work.     Figure 8 presents the RI and spot radius with different ROC. This result indicates that the optical performance can be further improved if using a curvature variable sensor.    Figure 10 shows the distortion characteristic of CZOS. Figure 11 shows the spot diagram of the systems using flat and curved image sensors. Figure 12 presents the comparison of optical performances between spherical lens system and aspherical lens system. Through the introduction of aspherical lenses, the optical performance (e.g., spot radius, relative illumination, field curvature, and MTF) can be further improved. Table 2 shows the used aspheric coefficient in Fig. 12.        Figure 13 presents the comparison of optical performances between CZOS and plastic based CZOS. The lenses of the CZOS can be replaced with a plastic material, as the performance of plastic based system (p-CZOS) is similar to the CZOS. Table 3 shows the plastic lens information of the system in Fig. 13.

Code, Data, and Materials Availability
No data were generated or analyzed in the presented research.