Ex) Article Title, Author, Keywords
Current Optics
and Photonics
Ex) Article Title, Author, Keywords
Curr. Opt. Photon. 2024; 8(5): 493-501
Published online October 25, 2024 https://doi.org/10.3807/COPP.2024.8.5.493
Copyright © Optical Society of Korea.
Hien Nguyen^{1}, Hieu Tran Doan Trung^{2,3}, Van Truong Vu^{1}, Hocheol Lee^{1}
Corresponding author: *hclee@hanbat.ac.kr, ORCID 0000-0001-7436-7567
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
This study proposes an effective visualization method for image distortion in high-resolution, machinable Fresnel mirrors, which offer significant advantages over traditional convex mirrors by being thinner and lighter. While commercial optical design programs are excellent at optimizing aberrations, they have some limitations in visualizing images from complex optical configurations. Therefore, NX^{TM} CAD software is employed to achieve photorealistic ray-traced visualization with high-fidelity image rendering due to its flexible two-dimensional and three-dimensional modeling environments. In comparative simulations with various mirror profiles, we identified an aspherical Fresnel mirror with a conic constant of k = −3 that can reduce distortion to 1.79%, according to Zemax OpticStudio^{®} calculations. Finally, the NX^{TM} software successfully validated the distortion image of our machinable aspherical Fresnel mirror design. Subsequent practical experiments validated the consistency between the predicted distortion and the actual visualization results. We anticipate that this specialized visualization technique holds the potential to radically transform the interactive design of optical systems that incorporate aspherical Fresnel mirrors.
Keywords: Aspherical surface, Conic constant, Convex mirror, Fresnel mirror, Image distortion
OCIS codes: (080.4228) Nonspherical mirror surfaces; (110.2960) Image analysis; (110.3000) Image quality assessment; (110.3010) Image reconstruction techniques; (220.2740) Geometric optical design
Convex spherical mirrors are a popular choice due to their wide viewing angles. Examples include rearview mirrors in cars and motorbikes [1], infrared mirrors for capturing radiation emitted by an absorber [2, 3], and dual Fresnel mirrors for three-dimensional (3D) displays [4]. However, their inherent curvature, defined by the radius of curvature (ROC), distorts images and reduces quality. Larger-diameter mirrors expand the field of view (FOV) but also increase distortion, which impairs the ability to accurately judge distances and positions in practical applications.
Several methods and mathematical functions are available for correcting this distortion error [5–9]. However, a physical solution is required for actual implementation to address this issue. Ideally, the mirror structure itself should minimize distortion without image processing. For instance, Jacob et al. [10] used spherical segments to create a large, aspheric primary mirror. Hasan et al. [11] proposed a method for visualizing the manufacturing tolerances of vehicle side mirrors by deviated curvature. Lee et al. [12] suggested a process for manufacturing an aspheric mirror using thin film, and an aspheric metal mirror was designed for use in the infrared region [13]. However, these methods are time-consuming and present practical implementation challenges. Consequently, this study introduces a simple and effective method for reducing image distortion that highlights the potential of the Fresnel structure to minimize such distortions.
A typical Fresnel structure comprises a series of concentric ridges or grooves etched onto a flat surface. Each ridge or groove functions as a small prism and bends light rays in a certain manner. In addition to their benefits of thinness, light weight, and effective light gathering, Fresnel structures have a wide range of applications owing to their distinctive characteristics. For example, Fresnel lenses are used to concentrate sunlight onto photovoltaic cells, which significantly improves solar power generation efficiency [14]. Furthermore, the Fresnel lens comprises micrometer-sized v-groove structures that control the maximum illuminance and brightness uniformity of LED−powered flashlights, which are used in high-quality photography [15], overhead projectors, and projection televisions [16]. They are also used in viewfinders on camera viewing systems and other optical devices [17, 18], including the Fresnel lens optical landing system, a visual landing assist that provides key glide slope information to pilots conducting carrier landings [19]. Considerable research has been conducted on the formation of Fresnel mirror images.
Our analysis begins with a detailed description of the mirror design. We then introduce Zemax OpticStudio^{®} (Ansys Inc., PA, USA)’s distortion calculation method, a well-established technique for rotationally symmetric mirrors. However, this method encounters some limitations when applied to the intricate configuration of our Fresnel mirror and its impact on image visualization. To address this, NX^{TM} CAD software (Siemens Digital Industries Software, TX, USA) is introduced in the following section. NX^{TM} offers a more effective approach to image rendering with its photorealistic ray-traced rendering capabilities. Next, we present practical experiments that serve to validate our simulations. Finally, we discuss the results and propose a novel Fresnel mirror design with significantly lower distortion and a lighter, thinner profile compared to traditional convex mirrors.
The most common form of a convex mirror has a rotationally symmetric surface, with the sag defined in Eq. (1):
where c is the base curvature at the vertex or the ROC of the spherical surface, k is a conic constant, and r is the radial coordinate of the point on the surface [20].
Mirror sag profiles obtained from Eq. (1) are shown in Fig. 1. They indicate that a lower conic constant k corresponds to a lower sag value and a flatter mirror shape. Therefore, an aspherical surface is expected to produce images with less distortion.
The geometric FOV α of each mirror can be determined using the Eq. (2):
where h_{max} is field height on axis (object height), d is the distance between the object and mirror, r is the radius of the mirror, R is Radius of curvature, and f is the focal length of the mirror.
Full FOV = θ_{wide} = 2α, meaning the object height varies linearly with the tangent of the field angle (see Fig. 2). The spherical convex mirror exhibited the largest FOV.
Equations (1) and (2) indicate that when the mirrors have identical indices, except for the conic constant, their sag and FOV will be different. In this situation, a lower conic constant results in a smaller FOV. Therefore, merely adjusting the conic constant for a conventional convex mirror may result in less distortion; however, it will also reduce the FOV of the mirror, resulting in a loss of corner visibility.
In this study, Fresnel mirrors were converted from convex mirrors while preserving the ROC and conic constants of the original surface. A Fresnel structure divides the surface into concentric circular sections called grooves, which function as microscopic convex mirrors that reflect light in a manner similar to a regular mirror. The angles and depths of these grooves were carefully designed to ensure that the light rays passing through them converge at a single virtual image point, mimicking the behavior of a regular mirror. Figure 3 illustrates two approaches to transforming a convex mirror into a Fresnel mirror. There are two main classifications of Fresnel mirrors: Constant pitch (p) and constant height (h). A constant-p-groove Fresnel mirror, shown in Fig. 3(b), divides the convex surface into grooves with equal spacing. Conversely, a constant-h-groove Fresnel mirror, shown in Fig. 3(c), features grooves with uniform depth. Importantly, each individual groove maintains the original ROC and conic constant of the surface in both designs. Thus, the generated Fresnel mirrors are flatter than convex mirrors, and their weight is significantly reduced.
As a reference for comparing the distortion and FOV specifications, the mirror configuration is based on practical examples, such as the conventional convex mirror used in convenience stores. The object size was selected to match the maximum FOV of the aspherical convex mirror and then scaled down for laboratory testing at a specific ratio. Table 1 details the specifications of a commercially available convex mirror for reference and configuration of Fresnel mirror systems used in this study.
TABLE 1 Mirrors specifications
Parameter | Spherical Mirror | Fresnel Mirror (Constant p) | Fresnel Mirror (Constant h) |
---|---|---|---|
Object Size (mm) | 800 × 800 | 800 × 800 | 800 × 800 |
Diameter_{mirror} (mm) | 50 | 50 | 50 |
ROC_{mirror} (mm) | 50 | 50 | 50 |
Distance between Object and Mirror (mm) | 400 | 400 | 400 |
Distance between Camera and Mirror (mm) | 200 | 200 | 200 |
First Pitch (mm) | - | 0.114 | 2.55 |
Minimum Pitch (mm) | - | 0.114 | 0.114 |
First Height (mm) | - | 0.00013 | 0.065 |
Maximum Height (mm) | - | 0.065 | 0.065 |
Zemax OpticStudio^{®} software simulates various optical surfaces, including rotationally symmetric spherical, aspherical, and Fresnel mirrors. When modeling Fresnel mirrors, this software simplifies modeling its own Fresnel surface profile by converting convex mirrors into Fresnel mirrors. It achieves this by transferring identical surface parameters, such as the ROC for each groove and mirror diameter, and treating their grooves as infinitesimally small (nanometer-sized). This allows the software to represent the mirror surface as a flat plane (Fig. 4) [21].
By directly computing the distortion change with variations in the conic constant of k, Zemax OpticStudio^{®} software was employed to estimate the distortion in its mirror design using specific parameters of diameter and ROC. This analysis allowed us to predict the maximum impact of modifying the conic constant on image distortion.
Grid distortion is used to display or compute the coordinates of the principal rays of a grid. In a distortion-free system, the chief ray coordinates of the image surface are linearly related to the field coordinates, as expressed in Eq. (3):
where x_{p} and y_{p} represent the predicted image coordinates relative to a reference image point, and f_{x} and f_{y} represent the linear coordinates of the object surface relative to a reference point. In optical systems where fields are defined by angles, f_{x} and f_{y} correspond to the tangents of these angles. Owing to the requirement of a linear relationship between the field coordinates and object/image positions, the tangents of the field angles are employed. To determine the ABCD matrix, Zemax OpticStudio^{®} employs ray tracing confined to a restricted region centered on a designated reference point within the field. The center of the FOV is typically selected as the reference, although Zemax OpticStudio^{®} provides flexibility in selecting an alternative reference location.
The Zemax OpticStudio^{®} program automatically positions the field grid corner in the object space at the maximum radial-field distance. For fields defined by angle, the entire field width is given by Eq. (4):
where θ_{r} represents the maximum radial field angle at the corner of the field.
The ray coordinates in the image space for an extremely narrow FOV were used to compute the ABCD matrix components. The ABCD matrix supports coordinate rotations. When the image surface is rotated such that the y object coordinate corresponds to both the x and y image coordinate, the ABCD matrix automatically accounts for the rotation. The grid distortion graphic shows the linear grid before revealing the real chief ray intercept for a ray with the same linear field coordinates, which is denoted by a symbol X at each grid point.
A rotationally symmetric optical system is required for typical radial distortion. Since distortion (P) is a vector, its magnitude is required to calculate the overall distortion.
The available text listing tabulates the expected image coordinate, the actual image coordinate, and the distortion in the Zemax OpticStudio^{®} program, defined as [21]:
where
and
The subscripts r and p correspond to the real and predicted coordinates of the image surface, respectively, with respect to the reference field position and image location. Additionally, R_{real} is the real image radius and R_{predicted} is the predicted image radius. Because both R_{real} and R_{predicted} are always positive, this definition will always result in a positive value for P. However, distinguishing between positive and negative distortions is still a frequently used term. To illustrate this, if R_{real} is less than R_{predicted}, the sign of P is altered to negative, a phenomenon known as negative distortion or barrel distortion as shown in Fig. 5(a). In contrast, a positive value for P is referred to as positive distortion or pincushion distortion as presented in Fig. 5(b).
The difference in distortion between Fresnel mirrors and convex mirrors is mainly due to the difference in sag, which leads to the optical path difference. Given the same ROC, diameter, and conic constant, Fresnel mirrors are significantly thinner than convex mirrors.
While commercial optical design programs excel at aberration optimization, they show some limitations in visualizing optical configurations. Additionally, their visualization functions often suffer from defocusing and inversion [22], making it difficult to accurately assess image formation by the mirrors. For our machinable Fresnel profiles, we switched to the NX^{TM} program to achieve a more efficient visualization process.
In this study, photorealistic ray tracing was employed to generate high-fidelity images of each mirror model. This established optical approach, documented in prior research [23, 24], simulates real-world photography [11, 25] and facilitates the creation of 3D models [26, 27]. Building on the concept presented by Tran et al. [22], who used NX^{TM} software to visualize freeform reflection surfaces, this work adopted the same software to render images for each mirror. These rendered images depict real-life objects reflected by the mirrors, allowing for an on-axis visualization to assess the performance of each mirror.
The photorealistic ray-traced visualization process followed a sequential approach, as illustrated in Fig. 6. The first step involved collecting point coordinates that define the mirror surface. This was achieved by building a simple calculation algorithm in MATLAB^{®} software based on Eq. (1), which generated specific numerical values representing the coordinates of surface points along the mirror cross-section, as shown in Fig. 7(a). These surface parameters were then imported into NX^{TM} software for the straightforward creation of 2D mirror models. Subsequently, the 2D model was transformed into a 3D mirror model. Figure 7(b) shows the general 3D models of the convex and Fresnel mirrors, respectively.
To define the FOV in front of the mirrors, an 800 mm × 800 mm line grid was positioned approximately 400 mm from the mirror surface. The final step in setting up the photorealistic ray-traced visualization simulation experiment involved positioning a camera on the axis in front of the mirror. This placement ensured that the camera captured the reflected image directly between the mirror and the grid. Following the completion of the setup process, Fig. 8 depicts the final configuration of the simulated experimental system.
Examination of both spherical and aspherical convex mirrors revealed a high degree of similarity in their overall shapes, primarily due to the negligible difference in their sag, which is a key parameter defining the curvature of the reflective surface. Similarly, Fresnel mirrors, characterized by their flat substrates, mainly exhibit variations in their groove structures and conic constants, but these differences have minimal impact on their overall shapes.
This experiment assessed the reliability of our photorealistic visualization method by comparing its predictions to the actual distortion produced by mirrors. We used a straightforward setup to realistically evaluate distortion. A small convex mirror served as a reference element for image distortion, while an aspherical groove Fresnel lens was directly attached to a flat mirror to mimic a Fresnel mirror (Fig. 9). The experimental setup mirrored the visualization process for consistency. The specific parameters of these mirrors, based on the specifications of commercially available convex mirrors (Edmund Optics Co., NJ, USA), are detailed in Table 2.
TABLE 2 Specifications for practical extension experiments
Parameter | Spherical Mirror | Fresnel Lens |
---|---|---|
Object Size (mm) | 800 × 800 | 800 × 800 |
Diameter_{mirror} (mm) | 33 | 33 |
ROC_{mirror} (mm) | 50 | 12.5 |
Focal Length (mm) | 25 | 25.04 |
Distance between Object and Mirror (mm) | 735 | 735 |
Distance between Camera and Mirror (mm) | 350 | 350 |
Groove Density (ea/inch) | - | 200 |
Conic Constant | 0 | −1 |
Figure 10 presents a distortion analysis conducted using Zemax OpticStudio^{®} for mirrors with conic constants ranging from 0 to – 6. The analysis reveals a key distinction between traditional convex and Fresnel mirrors. While convex mirrors exhibit a decrease in absolute distortion with a more negative conic constant, this is not the case for the Fresnel mirrors. Notably, the Fresnel mirror achieves its minimum distortion value at a conic constant of k = −3, a mere 1.79% distortion. This value is significantly lower than the distortion observed in both spherical convex mirrors and the maximum allowable distortion of 5% for lateral-view mirrors [28]. Since P is positive, the resulting image exhibits pincushion distortion. In comparison, convex mirrors with the same conic constant k = −3 exhibit a much higher distortion of around 15%. Our theoretical analysis also predicts that Fresnel mirrors with grooves characterized by an aspherical surface with a conic constant of k = −3 will exhibit minimal image distortion compared to mirrors with other conic constant values.
After evaluating distortion in the mirrors within Zemax OpticStudio^{®}, we selected three specific conic constants (k = 0, −1, and −3) to guide the design of a Fresnel mirror, considering the limitations of common standard manufacturing equipment. These constants correspond to a spherical-groove mirror, a parabolic-groove mirror, and a hyperbolic-groove mirror, respectively.
The results of the photorealistic ray-traced visualization generated by the NX^{TM} program are presented in Fig. 11. Owing to the resolution of the device, aspherical surfaces induce waviness in the image; However, this has no significant impact in practice. The white-and-black background is a general rendering characteristic in NX^{TM} and not related to graphical distortion. Notably, the Fresnel mirror with a conic constant of k = −3 shows an image with very small distortion. Furthermore, compared to a convex mirror with the same conic constant, the Fresnel mirror offers a similar FOV with significantly reduced distortion. In essence, the visualization results demonstrate that aspherical mirrors and aspherical-groove Fresnel mirrors outperform convex spherical mirrors in minimizing distortion while maintaining a comparable FOV. Additionally, the observed distortion in the reflected images aligns closely with the predictions from the Zemax OpticStudio^{®} simulation. The NX^{TM} software has successfully visualized images produced by microgroove Fresnel mirrors, facilitating the observation and evaluation of how the mirror’s surface elements influence the resulting image.
Beyond calculating distortion for the parameters listed in Table 2 using Zemax OpticStudio^{®}, the software was also employed to predict the distortion of actual products at various conic constants. The results of these predictions are presented in Fig. 12. Although the combination of a microgroove Fresnel lens (k = −1) and a flat mirror exhibits significant distortion, exceeding 34% − substantially higher than conventional convex mirrors − theoretical calculations predict minimal distortion for this system when the conic constant of the Fresnel lens approaches k = −2.
Figure 13 presents the results of the actual experiments comparing the distortion of a spherical convex mirror to a system designed to mimic a Fresnel mirror with the parameters outlined in Table 2. The observed distortion patterns align closely with those predicted by Zemax OpticStudio^{®}. External factors such as the light source and the transmission properties of the material used are the main reason for the blur observed in Fig. 13. However, these factors do not affect the optical distortion of the generated images. In essence, the data confirmed a consistent trend in distortion with three methods: Theoretical calculations using Zemax OpticStudio^{®}, photorealistic ray-traced visualization in NX^{TM} software, and actual image distortion analysis. This consistency validates the effectiveness of the NX^{TM} visualization approach. The analysis demonstrates that the photorealistic ray-traced visualization method employed in NX^{TM} provides a more intuitive approach to image quality evaluation compared to conventional commercial optical design software. The NX^{TM} visualization allows for direct observation of the final image and the impact of design changes. Moreover, this practical experiment confirms that the aspherical groove Fresnel lens and flat mirror system offer a significantly wider FOV compared to a conventional convex mirror. As illustrated in Fig. 13, this system not only reflects the entire object but also captures a broad area behind it, whereas the convex mirror only provides an image of the object.
Recognizing the impact of mirror distortion and limitations of commercial optical design software such as Zemax OpticStudio^{®}, this study successfully exploited the potential of photorealistic ray-tracing visualization to optimize the design of low-distortion aspherical-groove Fresnel mirrors. Our approach leveraged the real-time visualization capabilities of NX^{TM} software to directly observe and analyze distortion during the design process. We identified the optimal design with simulations comparing conventional convex mirrors to aspherical-groove Fresnel mirrors with varying conic constants. Specifically, we found that the Fresnel mirror with a conic constant of k = −3 exhibited a notably low distortion level, predicted to be 1.79% according to Zemax OpticStudio^{®} calculations. This represents a significant improvement over traditional convex mirrors while maintaining a comparable FOV. These findings demonstrate the effectiveness of photorealistic ray tracing as a visualization tool for designing low-distortion aspherical-groove Fresnel mirrors and offer a straightforward and practical solution for applications where a large FOV and accurate spatial judgment are critical.
We validated our visualization method with a practical experiment using commercially available components. The results showed a high similarity between predicted distortion and the actual image produced by a combination of an aspherical-groove Fresnel lens with a flat mirror. Furthermore, we explored the potential of using flexible Fresnel lenses. This approach shows promise for the development of innovative mirror devices with a broader FOV while maintaining low distortion. These advancements can be achieved without the need for complex, regulated manufacturing processes.
Based on the success of this study, we aim to continuously leverage this method to optimize designs for the fabrication process that leads to product actualization, such as the optimization of Fresnel mirror system designs that require both very low distortion and a wide FOV.
This study was supported by a National Research Foundation of Korea (NRF) grant funded by the Korean Government (Grant no. NRF-2022R1A2B5B01002532).
The authors declare that they have no competing financial interests or personal relationships that may have influenced the work reported in this study.
The 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.
Curr. Opt. Photon. 2024; 8(5): 493-501
Published online October 25, 2024 https://doi.org/10.3807/COPP.2024.8.5.493
Copyright © Optical Society of Korea.
Hien Nguyen^{1}, Hieu Tran Doan Trung^{2,3}, Van Truong Vu^{1}, Hocheol Lee^{1}
^{1}Department of Mechanical Engineering, Hanbat National University, Daejeon 34158, Korea
^{2}Department of Science of Measurement, University of Science and Technology, Daejeon 34113, Korea
^{3}Optical Imaging and Metrology Team, Advanced Instrumentation Institute, Korea Research Institute of Standards and Science, Daejeon 34113, Korea
Correspondence to:*hclee@hanbat.ac.kr, ORCID 0000-0001-7436-7567
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
This study proposes an effective visualization method for image distortion in high-resolution, machinable Fresnel mirrors, which offer significant advantages over traditional convex mirrors by being thinner and lighter. While commercial optical design programs are excellent at optimizing aberrations, they have some limitations in visualizing images from complex optical configurations. Therefore, NX^{TM} CAD software is employed to achieve photorealistic ray-traced visualization with high-fidelity image rendering due to its flexible two-dimensional and three-dimensional modeling environments. In comparative simulations with various mirror profiles, we identified an aspherical Fresnel mirror with a conic constant of k = −3 that can reduce distortion to 1.79%, according to Zemax OpticStudio^{®} calculations. Finally, the NX^{TM} software successfully validated the distortion image of our machinable aspherical Fresnel mirror design. Subsequent practical experiments validated the consistency between the predicted distortion and the actual visualization results. We anticipate that this specialized visualization technique holds the potential to radically transform the interactive design of optical systems that incorporate aspherical Fresnel mirrors.
Keywords: Aspherical surface, Conic constant, Convex mirror, Fresnel mirror, Image distortion
Convex spherical mirrors are a popular choice due to their wide viewing angles. Examples include rearview mirrors in cars and motorbikes [1], infrared mirrors for capturing radiation emitted by an absorber [2, 3], and dual Fresnel mirrors for three-dimensional (3D) displays [4]. However, their inherent curvature, defined by the radius of curvature (ROC), distorts images and reduces quality. Larger-diameter mirrors expand the field of view (FOV) but also increase distortion, which impairs the ability to accurately judge distances and positions in practical applications.
Several methods and mathematical functions are available for correcting this distortion error [5–9]. However, a physical solution is required for actual implementation to address this issue. Ideally, the mirror structure itself should minimize distortion without image processing. For instance, Jacob et al. [10] used spherical segments to create a large, aspheric primary mirror. Hasan et al. [11] proposed a method for visualizing the manufacturing tolerances of vehicle side mirrors by deviated curvature. Lee et al. [12] suggested a process for manufacturing an aspheric mirror using thin film, and an aspheric metal mirror was designed for use in the infrared region [13]. However, these methods are time-consuming and present practical implementation challenges. Consequently, this study introduces a simple and effective method for reducing image distortion that highlights the potential of the Fresnel structure to minimize such distortions.
A typical Fresnel structure comprises a series of concentric ridges or grooves etched onto a flat surface. Each ridge or groove functions as a small prism and bends light rays in a certain manner. In addition to their benefits of thinness, light weight, and effective light gathering, Fresnel structures have a wide range of applications owing to their distinctive characteristics. For example, Fresnel lenses are used to concentrate sunlight onto photovoltaic cells, which significantly improves solar power generation efficiency [14]. Furthermore, the Fresnel lens comprises micrometer-sized v-groove structures that control the maximum illuminance and brightness uniformity of LED−powered flashlights, which are used in high-quality photography [15], overhead projectors, and projection televisions [16]. They are also used in viewfinders on camera viewing systems and other optical devices [17, 18], including the Fresnel lens optical landing system, a visual landing assist that provides key glide slope information to pilots conducting carrier landings [19]. Considerable research has been conducted on the formation of Fresnel mirror images.
Our analysis begins with a detailed description of the mirror design. We then introduce Zemax OpticStudio^{®} (Ansys Inc., PA, USA)’s distortion calculation method, a well-established technique for rotationally symmetric mirrors. However, this method encounters some limitations when applied to the intricate configuration of our Fresnel mirror and its impact on image visualization. To address this, NX^{TM} CAD software (Siemens Digital Industries Software, TX, USA) is introduced in the following section. NX^{TM} offers a more effective approach to image rendering with its photorealistic ray-traced rendering capabilities. Next, we present practical experiments that serve to validate our simulations. Finally, we discuss the results and propose a novel Fresnel mirror design with significantly lower distortion and a lighter, thinner profile compared to traditional convex mirrors.
The most common form of a convex mirror has a rotationally symmetric surface, with the sag defined in Eq. (1):
where c is the base curvature at the vertex or the ROC of the spherical surface, k is a conic constant, and r is the radial coordinate of the point on the surface [20].
Mirror sag profiles obtained from Eq. (1) are shown in Fig. 1. They indicate that a lower conic constant k corresponds to a lower sag value and a flatter mirror shape. Therefore, an aspherical surface is expected to produce images with less distortion.
The geometric FOV α of each mirror can be determined using the Eq. (2):
where h_{max} is field height on axis (object height), d is the distance between the object and mirror, r is the radius of the mirror, R is Radius of curvature, and f is the focal length of the mirror.
Full FOV = θ_{wide} = 2α, meaning the object height varies linearly with the tangent of the field angle (see Fig. 2). The spherical convex mirror exhibited the largest FOV.
Equations (1) and (2) indicate that when the mirrors have identical indices, except for the conic constant, their sag and FOV will be different. In this situation, a lower conic constant results in a smaller FOV. Therefore, merely adjusting the conic constant for a conventional convex mirror may result in less distortion; however, it will also reduce the FOV of the mirror, resulting in a loss of corner visibility.
In this study, Fresnel mirrors were converted from convex mirrors while preserving the ROC and conic constants of the original surface. A Fresnel structure divides the surface into concentric circular sections called grooves, which function as microscopic convex mirrors that reflect light in a manner similar to a regular mirror. The angles and depths of these grooves were carefully designed to ensure that the light rays passing through them converge at a single virtual image point, mimicking the behavior of a regular mirror. Figure 3 illustrates two approaches to transforming a convex mirror into a Fresnel mirror. There are two main classifications of Fresnel mirrors: Constant pitch (p) and constant height (h). A constant-p-groove Fresnel mirror, shown in Fig. 3(b), divides the convex surface into grooves with equal spacing. Conversely, a constant-h-groove Fresnel mirror, shown in Fig. 3(c), features grooves with uniform depth. Importantly, each individual groove maintains the original ROC and conic constant of the surface in both designs. Thus, the generated Fresnel mirrors are flatter than convex mirrors, and their weight is significantly reduced.
As a reference for comparing the distortion and FOV specifications, the mirror configuration is based on practical examples, such as the conventional convex mirror used in convenience stores. The object size was selected to match the maximum FOV of the aspherical convex mirror and then scaled down for laboratory testing at a specific ratio. Table 1 details the specifications of a commercially available convex mirror for reference and configuration of Fresnel mirror systems used in this study.
TABLE 1. Mirrors specifications.
Parameter | Spherical Mirror | Fresnel Mirror (Constant p) | Fresnel Mirror (Constant h) |
---|---|---|---|
Object Size (mm) | 800 × 800 | 800 × 800 | 800 × 800 |
Diameter_{mirror} (mm) | 50 | 50 | 50 |
ROC_{mirror} (mm) | 50 | 50 | 50 |
Distance between Object and Mirror (mm) | 400 | 400 | 400 |
Distance between Camera and Mirror (mm) | 200 | 200 | 200 |
First Pitch (mm) | - | 0.114 | 2.55 |
Minimum Pitch (mm) | - | 0.114 | 0.114 |
First Height (mm) | - | 0.00013 | 0.065 |
Maximum Height (mm) | - | 0.065 | 0.065 |
Zemax OpticStudio^{®} software simulates various optical surfaces, including rotationally symmetric spherical, aspherical, and Fresnel mirrors. When modeling Fresnel mirrors, this software simplifies modeling its own Fresnel surface profile by converting convex mirrors into Fresnel mirrors. It achieves this by transferring identical surface parameters, such as the ROC for each groove and mirror diameter, and treating their grooves as infinitesimally small (nanometer-sized). This allows the software to represent the mirror surface as a flat plane (Fig. 4) [21].
By directly computing the distortion change with variations in the conic constant of k, Zemax OpticStudio^{®} software was employed to estimate the distortion in its mirror design using specific parameters of diameter and ROC. This analysis allowed us to predict the maximum impact of modifying the conic constant on image distortion.
Grid distortion is used to display or compute the coordinates of the principal rays of a grid. In a distortion-free system, the chief ray coordinates of the image surface are linearly related to the field coordinates, as expressed in Eq. (3):
where x_{p} and y_{p} represent the predicted image coordinates relative to a reference image point, and f_{x} and f_{y} represent the linear coordinates of the object surface relative to a reference point. In optical systems where fields are defined by angles, f_{x} and f_{y} correspond to the tangents of these angles. Owing to the requirement of a linear relationship between the field coordinates and object/image positions, the tangents of the field angles are employed. To determine the ABCD matrix, Zemax OpticStudio^{®} employs ray tracing confined to a restricted region centered on a designated reference point within the field. The center of the FOV is typically selected as the reference, although Zemax OpticStudio^{®} provides flexibility in selecting an alternative reference location.
The Zemax OpticStudio^{®} program automatically positions the field grid corner in the object space at the maximum radial-field distance. For fields defined by angle, the entire field width is given by Eq. (4):
where θ_{r} represents the maximum radial field angle at the corner of the field.
The ray coordinates in the image space for an extremely narrow FOV were used to compute the ABCD matrix components. The ABCD matrix supports coordinate rotations. When the image surface is rotated such that the y object coordinate corresponds to both the x and y image coordinate, the ABCD matrix automatically accounts for the rotation. The grid distortion graphic shows the linear grid before revealing the real chief ray intercept for a ray with the same linear field coordinates, which is denoted by a symbol X at each grid point.
A rotationally symmetric optical system is required for typical radial distortion. Since distortion (P) is a vector, its magnitude is required to calculate the overall distortion.
The available text listing tabulates the expected image coordinate, the actual image coordinate, and the distortion in the Zemax OpticStudio^{®} program, defined as [21]:
where
and
The subscripts r and p correspond to the real and predicted coordinates of the image surface, respectively, with respect to the reference field position and image location. Additionally, R_{real} is the real image radius and R_{predicted} is the predicted image radius. Because both R_{real} and R_{predicted} are always positive, this definition will always result in a positive value for P. However, distinguishing between positive and negative distortions is still a frequently used term. To illustrate this, if R_{real} is less than R_{predicted}, the sign of P is altered to negative, a phenomenon known as negative distortion or barrel distortion as shown in Fig. 5(a). In contrast, a positive value for P is referred to as positive distortion or pincushion distortion as presented in Fig. 5(b).
The difference in distortion between Fresnel mirrors and convex mirrors is mainly due to the difference in sag, which leads to the optical path difference. Given the same ROC, diameter, and conic constant, Fresnel mirrors are significantly thinner than convex mirrors.
While commercial optical design programs excel at aberration optimization, they show some limitations in visualizing optical configurations. Additionally, their visualization functions often suffer from defocusing and inversion [22], making it difficult to accurately assess image formation by the mirrors. For our machinable Fresnel profiles, we switched to the NX^{TM} program to achieve a more efficient visualization process.
In this study, photorealistic ray tracing was employed to generate high-fidelity images of each mirror model. This established optical approach, documented in prior research [23, 24], simulates real-world photography [11, 25] and facilitates the creation of 3D models [26, 27]. Building on the concept presented by Tran et al. [22], who used NX^{TM} software to visualize freeform reflection surfaces, this work adopted the same software to render images for each mirror. These rendered images depict real-life objects reflected by the mirrors, allowing for an on-axis visualization to assess the performance of each mirror.
The photorealistic ray-traced visualization process followed a sequential approach, as illustrated in Fig. 6. The first step involved collecting point coordinates that define the mirror surface. This was achieved by building a simple calculation algorithm in MATLAB^{®} software based on Eq. (1), which generated specific numerical values representing the coordinates of surface points along the mirror cross-section, as shown in Fig. 7(a). These surface parameters were then imported into NX^{TM} software for the straightforward creation of 2D mirror models. Subsequently, the 2D model was transformed into a 3D mirror model. Figure 7(b) shows the general 3D models of the convex and Fresnel mirrors, respectively.
To define the FOV in front of the mirrors, an 800 mm × 800 mm line grid was positioned approximately 400 mm from the mirror surface. The final step in setting up the photorealistic ray-traced visualization simulation experiment involved positioning a camera on the axis in front of the mirror. This placement ensured that the camera captured the reflected image directly between the mirror and the grid. Following the completion of the setup process, Fig. 8 depicts the final configuration of the simulated experimental system.
Examination of both spherical and aspherical convex mirrors revealed a high degree of similarity in their overall shapes, primarily due to the negligible difference in their sag, which is a key parameter defining the curvature of the reflective surface. Similarly, Fresnel mirrors, characterized by their flat substrates, mainly exhibit variations in their groove structures and conic constants, but these differences have minimal impact on their overall shapes.
This experiment assessed the reliability of our photorealistic visualization method by comparing its predictions to the actual distortion produced by mirrors. We used a straightforward setup to realistically evaluate distortion. A small convex mirror served as a reference element for image distortion, while an aspherical groove Fresnel lens was directly attached to a flat mirror to mimic a Fresnel mirror (Fig. 9). The experimental setup mirrored the visualization process for consistency. The specific parameters of these mirrors, based on the specifications of commercially available convex mirrors (Edmund Optics Co., NJ, USA), are detailed in Table 2.
TABLE 2. Specifications for practical extension experiments.
Parameter | Spherical Mirror | Fresnel Lens |
---|---|---|
Object Size (mm) | 800 × 800 | 800 × 800 |
Diameter_{mirror} (mm) | 33 | 33 |
ROC_{mirror} (mm) | 50 | 12.5 |
Focal Length (mm) | 25 | 25.04 |
Distance between Object and Mirror (mm) | 735 | 735 |
Distance between Camera and Mirror (mm) | 350 | 350 |
Groove Density (ea/inch) | - | 200 |
Conic Constant | 0 | −1 |
Figure 10 presents a distortion analysis conducted using Zemax OpticStudio^{®} for mirrors with conic constants ranging from 0 to – 6. The analysis reveals a key distinction between traditional convex and Fresnel mirrors. While convex mirrors exhibit a decrease in absolute distortion with a more negative conic constant, this is not the case for the Fresnel mirrors. Notably, the Fresnel mirror achieves its minimum distortion value at a conic constant of k = −3, a mere 1.79% distortion. This value is significantly lower than the distortion observed in both spherical convex mirrors and the maximum allowable distortion of 5% for lateral-view mirrors [28]. Since P is positive, the resulting image exhibits pincushion distortion. In comparison, convex mirrors with the same conic constant k = −3 exhibit a much higher distortion of around 15%. Our theoretical analysis also predicts that Fresnel mirrors with grooves characterized by an aspherical surface with a conic constant of k = −3 will exhibit minimal image distortion compared to mirrors with other conic constant values.
After evaluating distortion in the mirrors within Zemax OpticStudio^{®}, we selected three specific conic constants (k = 0, −1, and −3) to guide the design of a Fresnel mirror, considering the limitations of common standard manufacturing equipment. These constants correspond to a spherical-groove mirror, a parabolic-groove mirror, and a hyperbolic-groove mirror, respectively.
The results of the photorealistic ray-traced visualization generated by the NX^{TM} program are presented in Fig. 11. Owing to the resolution of the device, aspherical surfaces induce waviness in the image; However, this has no significant impact in practice. The white-and-black background is a general rendering characteristic in NX^{TM} and not related to graphical distortion. Notably, the Fresnel mirror with a conic constant of k = −3 shows an image with very small distortion. Furthermore, compared to a convex mirror with the same conic constant, the Fresnel mirror offers a similar FOV with significantly reduced distortion. In essence, the visualization results demonstrate that aspherical mirrors and aspherical-groove Fresnel mirrors outperform convex spherical mirrors in minimizing distortion while maintaining a comparable FOV. Additionally, the observed distortion in the reflected images aligns closely with the predictions from the Zemax OpticStudio^{®} simulation. The NX^{TM} software has successfully visualized images produced by microgroove Fresnel mirrors, facilitating the observation and evaluation of how the mirror’s surface elements influence the resulting image.
Beyond calculating distortion for the parameters listed in Table 2 using Zemax OpticStudio^{®}, the software was also employed to predict the distortion of actual products at various conic constants. The results of these predictions are presented in Fig. 12. Although the combination of a microgroove Fresnel lens (k = −1) and a flat mirror exhibits significant distortion, exceeding 34% − substantially higher than conventional convex mirrors − theoretical calculations predict minimal distortion for this system when the conic constant of the Fresnel lens approaches k = −2.
Figure 13 presents the results of the actual experiments comparing the distortion of a spherical convex mirror to a system designed to mimic a Fresnel mirror with the parameters outlined in Table 2. The observed distortion patterns align closely with those predicted by Zemax OpticStudio^{®}. External factors such as the light source and the transmission properties of the material used are the main reason for the blur observed in Fig. 13. However, these factors do not affect the optical distortion of the generated images. In essence, the data confirmed a consistent trend in distortion with three methods: Theoretical calculations using Zemax OpticStudio^{®}, photorealistic ray-traced visualization in NX^{TM} software, and actual image distortion analysis. This consistency validates the effectiveness of the NX^{TM} visualization approach. The analysis demonstrates that the photorealistic ray-traced visualization method employed in NX^{TM} provides a more intuitive approach to image quality evaluation compared to conventional commercial optical design software. The NX^{TM} visualization allows for direct observation of the final image and the impact of design changes. Moreover, this practical experiment confirms that the aspherical groove Fresnel lens and flat mirror system offer a significantly wider FOV compared to a conventional convex mirror. As illustrated in Fig. 13, this system not only reflects the entire object but also captures a broad area behind it, whereas the convex mirror only provides an image of the object.
Recognizing the impact of mirror distortion and limitations of commercial optical design software such as Zemax OpticStudio^{®}, this study successfully exploited the potential of photorealistic ray-tracing visualization to optimize the design of low-distortion aspherical-groove Fresnel mirrors. Our approach leveraged the real-time visualization capabilities of NX^{TM} software to directly observe and analyze distortion during the design process. We identified the optimal design with simulations comparing conventional convex mirrors to aspherical-groove Fresnel mirrors with varying conic constants. Specifically, we found that the Fresnel mirror with a conic constant of k = −3 exhibited a notably low distortion level, predicted to be 1.79% according to Zemax OpticStudio^{®} calculations. This represents a significant improvement over traditional convex mirrors while maintaining a comparable FOV. These findings demonstrate the effectiveness of photorealistic ray tracing as a visualization tool for designing low-distortion aspherical-groove Fresnel mirrors and offer a straightforward and practical solution for applications where a large FOV and accurate spatial judgment are critical.
We validated our visualization method with a practical experiment using commercially available components. The results showed a high similarity between predicted distortion and the actual image produced by a combination of an aspherical-groove Fresnel lens with a flat mirror. Furthermore, we explored the potential of using flexible Fresnel lenses. This approach shows promise for the development of innovative mirror devices with a broader FOV while maintaining low distortion. These advancements can be achieved without the need for complex, regulated manufacturing processes.
Based on the success of this study, we aim to continuously leverage this method to optimize designs for the fabrication process that leads to product actualization, such as the optimization of Fresnel mirror system designs that require both very low distortion and a wide FOV.
This study was supported by a National Research Foundation of Korea (NRF) grant funded by the Korean Government (Grant no. NRF-2022R1A2B5B01002532).
The authors declare that they have no competing financial interests or personal relationships that may have influenced the work reported in this study.
The 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.
TABLE 1 Mirrors specifications
Parameter | Spherical Mirror | Fresnel Mirror (Constant p) | Fresnel Mirror (Constant h) |
---|---|---|---|
Object Size (mm) | 800 × 800 | 800 × 800 | 800 × 800 |
Diameter_{mirror} (mm) | 50 | 50 | 50 |
ROC_{mirror} (mm) | 50 | 50 | 50 |
Distance between Object and Mirror (mm) | 400 | 400 | 400 |
Distance between Camera and Mirror (mm) | 200 | 200 | 200 |
First Pitch (mm) | - | 0.114 | 2.55 |
Minimum Pitch (mm) | - | 0.114 | 0.114 |
First Height (mm) | - | 0.00013 | 0.065 |
Maximum Height (mm) | - | 0.065 | 0.065 |
TABLE 2 Specifications for practical extension experiments
Parameter | Spherical Mirror | Fresnel Lens |
---|---|---|
Object Size (mm) | 800 × 800 | 800 × 800 |
Diameter_{mirror} (mm) | 33 | 33 |
ROC_{mirror} (mm) | 50 | 12.5 |
Focal Length (mm) | 25 | 25.04 |
Distance between Object and Mirror (mm) | 735 | 735 |
Distance between Camera and Mirror (mm) | 350 | 350 |
Groove Density (ea/inch) | - | 200 |
Conic Constant | 0 | −1 |