
- Chinese Optics Letters
- Vol. 21, Issue 2, 023601 (2023)
Abstract
Keywords
1. Introduction
Wide field-of-view (WFOV) optics, exemplified by fisheye lenses featuring a field-of-view (FOV) close to or even exceeding 180°, are widely employed in landscape photography, security surveillance, meteorology, and image projection[1–4]. In recent years, they are also starting to gain traction in emerging electronics and optics products, enabling panoramic cameras and 3D depth sensors, augmented reality/virtual reality (AR/VR) optics rendering immersive experiences, omnidirectional computer vision systems, and new biomedical imaging instruments. To fulfill these application demands, suppression of off-axis optical aberrations such as coma, astigmatism, and field curvature is crucial to realizing high-quality WFOV optics. The traditional approach for aberration mitigation entails cascading multiple lens elements, which, however, increases the size, weight, complexity, and cost of the optical system.
Flat optics based on optical metasurfaces or diffractive optical elements (DOEs) offer an alternative solution to expand the FOV of optical systems. One scheme involves stacking multiple metasurfaces, and diffraction-limited FOVs up to 56° have been attained using this method[5–10]. Combining a single-layer metasurface or diffractive lens with a physical or virtual optical aperture provides an architecturally simpler approach[11–22]. In particular, a single-element fisheye metalens was demonstrated with
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In this paper, we derive an analytical solution to the optimum phase profile of a WFOV flat lens assuming the single-layer geometry, yielding results in excellent agreement with numerically optimized designs but without requiring computationally intensive optimization. The analytical solution is generically applicable to different operation wavelength ranges, lens/substrate materials, and meta-atom or diffractive element designs. Finally, we derive an expression relating design parameters with focusing performance and investigate the design trade-offs in realizing WFOV flat lenses.
2. Theoretical Model
The basic concept of the single-layer WFOV flat lens is illustrated in Fig. 1(a). An aperture is placed at the front surface of a substrate, and a metasurface (or a DOE surface) is patterned on the back surface to act as an optical phase mask. Beams from different angles of incidence (AOIs) are refracted at the front surface and arrive at different yet continuous portions of the backside phase mask. This architecture and optimized designs enable diffraction-limited focusing performance continuously across the near-180° FOV[23]. At large AOIs, the optical transmission drops due to the cosine dependence of the projected aperture area, which ultimately limits the practical FOV. This limitation can be potentially addressed by using a front aperture with a curved surface.
Figure 1.Schematic illustration of WFOV metalens design. (a) 3D structure. (b) Illustration of the phase profile derivation. (c) Illustration of the image height derivation. (a) is reprinted with permission from the American Chemical Society[
The phase profile of the metasurface will be derived by assuming stigmatic focusing for a pencil of parallel rays incident on the aperture from all directions across the 180° FOV. In the WFOV lens configuration depicted in Fig. 1(b), the phase profile of the metasurface is given by a function
Here,
The only unknown variable in Eq. (1) is
Since the two pencils of rays share the same metasurface area,
The AOI from free-space
Substituting Eq. (5) into Eq. (2) leads to the integral form of the target phase profile.
The derivation is generic and applicable to different wavelengths, substrate materials, and meta-atom or diffractive element designs. It can also be extended to cases with multiple substrate layers (with thickness of the
The main assumption in this analytical formalism is that
When
To ensure sharp focusing,
Denoting aperture diameter as
This expression explicitly reveals the dependence of lens performance on configuration parameters including focal length, aperture size, substrate thickness, and refractive index of the substrate. To achieve better performance, one can in general increase the
The equation reveals a similar dependence of lens performance on design parameters.
We show in the following that the design maintains diffraction-limited performance over the entire hemispherical FOV up to a moderate NA of
Figure 2.Calculated performance of an ideal WFOV lens. (a) Lens phase profile retrieved from analytical and numerical solutions. (b) Image heights with different AOIs from analytical and numerical solutions. The green dashed line represents the telecentric condition, which corresponds to
3. Comparison with Numerical Design
We then used the Kirchhoff diffraction integral[29] to evaluate the focusing performance of the lens. Assuming a meta-atom pitch of 4 µm, the lens focusing efficiency [defined as the fraction of power encircled within an area of a diameter equal to five times the focal spot full width at half-maximum (FWHM) normalized by the total incident power] and Strehl ratio as a function of AOI from air are shown in Fig. 2(c), and the focal spot profiles at several AOIs are displayed in Figs. 2(d)–2(g). The lens exhibits diffraction-limited focusing performance with Strehl ratios consistently larger than 0.8 and efficiencies higher than 75% over the entire hemispherical FOV.
The diffraction integral calculations above assume ideal meta-atoms, so the metasurface acts as a pure phase mask without imposing intensity modulation and phase error. To make a realistic estimate of the metalens efficiency, next, we incorporated actual meta-atom structures, and their optical characteristics were modeled using full-wave calculations[30]. The all-dielectric, freeform meta-atoms under consideration are made from 1 µm thick PbTe film resting on a
Figure 3.Simulated performance of a metalens composed of realistic meta-atoms. (a) Image height, (b) efficiency, and Strehl ratio for different AOIs based on full-wave modeled meta-atoms.
4. Impact of Design Parameters on Lens Performance
The analytical formalism allows computationally efficient design of WFOV flat lenses, especially in cases where ray-tracing-based numerical optimization cannot be implemented in a reasonable time scale. A comparison of the two design methods is presented in Appendix A. The analytical solution also elucidates the design trade-offs. For a given wavelength and substrate refractive index, the WFOV lens design is fully defined by three independent parameters: aperture size, substrate thickness, and focal length. In the following, we investigate the effect of varying aperture size, substrate thickness, and focal length on focusing performance of the lens for a substrate index
Figures 4(a) and 4(b) plot the focusing efficiency and Strehl ratio values (both averaged over the entire near-180° FOV) for WFOV flat lenses with varying NAs. In Fig. 4(a), the lens aperture diameter is fixed to 1 mm, the substrate thickness is 2 mm, and the focal length is varied to obtain different NAs. In Fig. 4(b), the focal length is set to 2 mm, the substrate thickness is 2 mm, and the aperture diameter is varied. Shorter focal length requires more abrupt change of optical phase, whereas spatial overlap between beams with different AOIs increases with larger aperture size, both of which negatively impact the focusing quality. Consequently, both efficiency and Strehl ratio decrease with increasing NA. Figure 4(c) depicts the impact of varying the substrate thickness. Increasing substrate thickness leads to lower spatial overlap between beams with different AOIs, thereby improving the focusing quality, albeit at the expense of larger device footprint, which explains the improvement of Strehl ratio at thicknesses less than 2 mm. Notably, when the substrate thickness exceeds 2 mm, the design significantly deviates from the telecentric configuration, resulting in lower Strehl ratios. As a result, an optimum thickness arises, which maximizes the Strehl ratio, as shown in Fig. 4(c). All these results are in accordance with Eq. (10).
Figure 4.(a), (b) Effects of NA on efficiency and Strehl ratio averaged over the entire near-180° FOV by changing (a) focal length and (b) aperture size. (c) Effects of substrate thickness on averaged efficiency and Strehl ratio.
5. Conclusion
In summary, we derived an analytical design approach for flat (metasurface or diffractive) fisheye lenses capable of imaging over near-180° FOV. We demonstrate that lenses designed using this scheme can achieve nearly diffraction-limited performance across the entire FOV while maintaining high focusing efficiencies above 65%. This design approach not only sheds light on the key design trade-offs of the WFOV lens, but is also poised to supersede the traditional iterative design scheme and significantly expedite deployment of the WFOV lens technology in diverse applications ranging from 3D sensing to biomedical imaging.
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