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Abstract
Metalenses are considered a promising solution for miniaturizing numerous optical systems due to their light weight, ultrathin thickness and compact size. However, it remains a challenge for metalenses to achieve both wide field-of-view and broadband achromatic imaging. In this work, a single-layer achromatic metalens with a wide field-of-view of 160° in the 3800 nm–4200 nm band is designed and analyzed. The quadratic phase profile of the metalens and the propagation phase of each meta-atom are used to increase the field-of-view and compensate for chromatic aberration, respectively. In addition, the metalens is capable of transverse achromatic imaging. The design can be extended to other optical frequencies, which is promising for applications in unmanned vehicles, infrared detection, etc.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Lenses are essential in various optical imaging systems including microscopes, telescopes and photographic projection systems. However, the demand for miniaturization, integration and multifunctionality in modern optical systems poses a challenge to conventional optical lenses because they are difficult to fabricate and integrate. To meet the specific requirements of applications such as smart wearables, smartphones and AR/VR glasses, metasurfaces have attracted much attention [1]. Metasurfaces are artificial nanostructured interfaces consisting of subwavelength meta-atoms. The phase, amplitude and polarization of light fields can be manipulated by appropriately designing the materials, sizes and shapes of the meta-atoms [2]. This outstanding capability has led to a lot of functional usages, such as anomalous reflectors [3,4], phase holography [5,6], optical sensing [7,8] and metalenses [9–11]. Among the various metasurfaces, metalenses are arguably the most fundamental and important [12]. Compared with conventional counterparts, metalenses offer unprecedented advantages of high integration, ultrathin thickness, compact structures and enhanced optical performance [13,14]. However, the presence of chromatic and off-axis aberrations hinders their widespread application.
Many methods have been proposed to develop achromatic metalenses that operate at specific discrete wavelengths [15–17] and broad bands [18,19], covering the visible [10,20], near-infrared [21] and mid-infrared regions [22,23]. One way to achieve achromatic imaging is to segment the aperture of the metalens to focus light beams of multiple discrete wavelengths in the same plane [24]. However, this method suffers from low efficiency and severe noise in practical situations. Another approach uses cascaded metalenses, which increases the difficulty of fabrication and integration [25]. For broadband achromatic imaging, one of the most common methods is based on dispersion engineering [20,26,27]. However, it should be noted that this method generally requires a large meta-atom database and the optimized structure of the meta-atom is usually quite complex.
Moreover, in practical applications, wide field-of-view imaging is highly attractive, but the existence of off-axis aberrations inherently restricts the field-of-view. Several solutions have been proposed to extend the field-of-view. One approach is to integrate the metalens on a curved substrate to eliminate the off-axis aberrations [28]. However, the metalens following this approach encounters difficulties related to weight and integration. The use of a double-layer cascaded metalens is a common strategy to realize a wide field-of-view [29–32]. In addition, the field-of-view, which is a function of the numerical aperture (NA), can theoretically be very wide with the proper design [33]. However, the fabrication of a double-layer metalens presents challenges in terms of precise alignment and assembly. Another approach is to introduce a shifted term in the phase profile of the metalens and integrate multiple metalenses into an array to attain wide-angle imaging at a given wavelength [34]. Metalenses designed in this way can have good imaging performance, but the fabrication cost is also increased. Luo et al. proposed a metalens with a quadratic phase profile to achieve a wide field-of-view [35]. Such a metalens can transform the rotational symmetry associated with oblique incidence into the translational symmetry of the focus. Martins et al. further analyzed the mechanism of this type of metalens using Fourier transforms [36]. In principle, the metalens with a quadratic phase profile can be approximated as an ideal spherical lens, and its refractive index and radius are infinite, allowing the correction of off-axis aberrations. Gu et al. designed a mid-infrared metalens with a field-of-view exceeding 170° at a wavelength of 5.2 µm by combining a quadratic phase profile with a stop aperture [37]. Li et al. proposed an achromatic wide-field metalens based on a quadratic phase profile with a field-of-view of 100° at harmonic wavelengths of 600 nm and 1200 nm [38]. Moreover, the aperture of the metalens is not constrained by the intrinsic material properties. Melati et al. analyzed the broadband behavior of a metalens with a quadratic phase profile [39]. The metalens has the ability to tolerate longitudinal and transverse chromatic aberrations in a 140 nm bandwidth. We summarize these typical metalenses with a wide field-of-view in Table 1.
Table 1. Typical wide field-of-view metalenses
Although different ways to extend the field-of-view have been reported, most can only work at certain wavelengths. Furthermore, most of the broadband achromatic wide field-of-view metalenses currently reported follow a complex double-layer cascade approach. Simultaneous wide field-of-view and achromatic imaging with a single-layer metalens has rarely been reported. In this work, a broadband achromatic wide field-of-view (BAWF) metalens combining quadratic phase profile and propagation phase compensation is proposed, which operates in the mid-infrared region from 3800 nm to 4200 nm with a field-of-view of 160°. Compared with diffraction-limited hyperbolic-phase achromatic (DLHA) and chromatic quadratic-phase (CQ) metalenses, the BAWF metalens is able to realize wide-angle and achromatic imaging, and the design can be extended to other optical frequencies. The single-layer BAWF metalens can promote the applications in unmanned vehicles, infrared detection and other related fields.
2. Principles
2.1 Phase modulation for wide field-of-view
In general, for normal incidence, the hyperbolic phase φH and quadratic phase φQ of a metalens are described by,
In Eqs. (1) and (2), k0 is the wave vector and k0 = 2π/λ0. λ0 is the operating wavelength, and next is the refractive index of the surrounding medium. f is the focal length, and r is the radial distance from the center of the metalens. According to the generalized Snell's law [2], we have,
Here dφ/dr is the phase gradient. nout and nin are the refractive indices of the two media in which the light is refracted and incident. In air, nout = nin =1. kr is the wave vector in the radial direction. θout and θin are the angles of refraction and incidence, respectively. From Eq. (3), the ratio kr/k0 characterizes the light deflection. Therefore, we evaluate the normalized wave vectors kcH and kcQ for the metalenses with hyperbolic and quadratic phase profiles at normal incidence in air, which are expressed by,
where krH and krQ are the radial wave vectors of the hyperbolic and quadratic phase profiles, respectively. From Eqs. (4) and (5), both kcH and kcQ are symmetric about r = 0. The two phase profiles possess the ability to symmetrically refract light from the center, resulting in effective focusing. When the light is incident at an angle θ, Eqs. (1) and (2) can be rewritten as, where φHO and φQO are the hyperbolic and quadratic phase profiles at oblique incidence, and θ is the incident angle. For hyperbolic and quadratic phase profiles, the corresponding ratios kcHO and kcQO at oblique incidence in air can be obtained as,Here krHO and krQO are the wave vectors at oblique incidence. According to Eqs. (8) and (9), kcHO loses symmetry with respect to the radial position r, while kcQO retains symmetry with the axis of symmetry shifted at r = fsinθ. Hence, in the case of oblique incidence, the refracted light of the metalens with a hyperbolic phase profile loses its radial symmetry and fails to focus, while the refracted light of the metalens with a quadratic phase profile still has symmetry and can achieve focusing. To demonstrate such differences, we simulated the performance of two metalenses with hyperbolic and quadratic phase profiles by using Eqs. (6)–(9) as shown in Fig. 1. In these ray-tracing simulations, the focal length was 25 µm, and the metalens diameter was 79.2 µm.
From Figs. 1(a) and 1(e), the metalens with a hyperbolic phase profile can achieve near-perfect focusing at normal incidence without spherical aberration, while the metalens with a quadratic phase profile can achieve focusing but with obvious spherical aberration. However, in practice, since the designed phase profile is discretized, the wavefront shaping will deviate from the ideal. As a result, the metalens with a hyperbolic phase profile still has spherical aberration. As shown in Figs. 1(b)–1(d), the refracted rays of the metalens with a quadratic phase profile retain a symmetric focusing ability at oblique incidence. On the contrary, the refracted rays of the metalens with a hyperbolic phase profile fail to focus perfectly as shown in Figs. 1(f)–1(h). These simulations confirm that the metalens with a quadratic phase profile enables wide-angle imaging.
Fig. 1. Deflection of the refracted light for the metalenses with quadratic and hyperbolic phase profiles, respectively. (a)–(d) Light deflection through the quadratic metalens at an incident angle θ of 0°, 10°, 30° and 50°, respectively. (e)–(f) Light deflection through the hyperbolic metalens at an incident angle θ of 0°, 10°, 30° and 50°, respectively.
2.2 Compensation for chromatic aberration
In general, for a working wavelength band in the range from λmin to λmax, the phase profile in Eq. (2) can be rewritten as,
Achromatic imaging can be realized when these relations of the fundamental phase φQ (r, λmax) and the compensation phase ΔφQ (r, λ0) are satisfied. Figure 2(a) illustrates the phase profile required for achromatic imaging in the mid-infrared where λmin and λmax are 3800 nm and 4200 nm, respectively, and the central operating wavelength λ0 is 4000 nm. Figure 2(b) shows the corresponding phases at different light frequencies and radial positions.
Fig. 2. Achromatic metalens. (a) Phase profiles for a BAWF metalens in a wavelength band ranging from λmin to λmax. Here λmin and λmax are 3800 nm and 4200 nm, respectively, and the central operating wavelength λ0 is 4000 nm. (b) Phases at different light frequencies and radial positions. The positions are taken from the vertical lines shown in (a).
The phase φQ (r, λmax) depends on λmax and can be modulated using the geometric phase associated with the rotation of the meta-atoms [40]. The geometric phase expresses the relationship between the rotation angle of the meta-atom and the phase modulation of the optical field. If the incident light is left-handed circularly polarized (LCP), the transmitted light can be expressed by the Jones vector [20],
The compensation phase ΔφQ (r, λ0) has a linear relationship with 1/λ0, which denotes the phase difference among different incident wavelengths and can be realized by the propagation phase [41,42]. However, the phase modulation of meta-atom does not necessarily vary linearly with 1/λ0. In the design, it is required to select specific meta-atoms that have a nearly linear ΔφQ versus 1/λ0 relationship to construct the metalens. The mechanism of the propagation phase is distinct from that of the geometric phase. Therefore, the two types of phases can be flexibly combined, and they will not interfere with each other.
3. Metalens design and simulation
In the design, the quadratic phase is formed as the fundamental phase, which is achieved by rotating the meta-atom. This fundamental phase is used to achieve focusing for light of wavelength λmax, which corresponds to φQ (r, λmax) in Eq. (10). The propagation phase is used to compensate the phase difference, which corresponds to Eq. (11). It is important to note that the meta-atoms will introduce an initial phase at λmax (see Supplement 1, Fig. S1). Therefore, the fundamental phase profile is actually achieved by both the geometric phase and the propagation phase. The metalens is realized by arranging an array of rectangular amorphous silicon (a-Si) meta-atoms on a SiO2 substrate, and the sketch of the meta-atom is shown in Fig. 3(a). The geometric phase can be regulated by rotating the meta-atom at an angle α, and the propagation phase can be controlled by changing the sizes namely the width W and the length L. The meta-atom period P is set to 1.8 µm, and the height H is kept at 3.5 µm. Here a-Si is employed as the composing dielectric material due to its high refractive index in the mid-infrared. This results in the confinement of the optical field within the meta-atom, and the coupling to the neighboring meta-atoms can be effectively reduced as shown in Fig. 3(c). We chose SiO2 as the substrate material because of its low refractive index, which avoids affecting the metasurface [22,43]. First, we used Lumerical finite difference time domain (FDTD) solution to rigorously calculate the phase delay and polarization conversion efficiency of the meta-atom. In these FDTD calculations, periodic boundary conditions were assumed along both the x- and y-directions. In addition, perfectly matched layer boundaries were considered in the z-direction. The metasurface was illuminated by LCP light with a wavelength range of 3800 nm–4200 nm. The meta-atom length L was swept from 800 nm to 1700nm in 31 steps, and the width W was swept from 200 nm to 800 nm in 46 steps.
Fig. 3. BAWF metalens design. (a) Schematic representation of the meta-atom. Period P, height H, length L, width W and rotation angle α are labeled. P and H are kept constant at 1.8 µm and 3.5 µm, respectively. (b) Phase modulation (i) and polarization conversion efficiency (ii) of each constituent meta-atom as a function of the spatial frequency of the incident light. (c) Normalized magnetic fields at different wavelengths for a meta-atom with width W of 0.72 µm and length L of 1.14 µm.
To achieve a strong linear relationship between the compensation phase and the frequency 1/λ0 as presented in Eq. (11), we selected meta-atoms with a linearity of at least 97% by least-squares fitting of the ΔφQ versus 1/λ0 curve. Meanwhile, the resulting propagation phase must satisfy the compensation phase. Eventually, we determined seventeen types of meta-atoms to construct the metalens (see Supplement 1, Table S1 for details). The respective phase and polarization conversion efficiency are shown in Fig. 3(b). We can find that the phase of each meta-atom has an almost linear dependence on the light frequency, which is meaningful for the chromatic compensation. Furthermore, the polarization conversion efficiency of most meta-atoms exceeds 20%. From Fig. 3(c), the multiple resonances inside the meta-atoms are the main contributors to the propagation phase modulation. After the candidate meta-atoms were determined, a BAWF metalens was constructed by spatially arranging them according to the desired phase profile and compensation phase. The metalens has a focal length f of 25 µm, a diameter D of 79.2 µm and an NA of 0.85. Here, the NA is calculated as NA = D/(D 2 + 4f 2)1/2. Simulations were further performed to study the focal fields at a range of wavelengths and incident angles (see Supplement 1, Figs. S2–S6). In short, all these results show that the BAWF metalens can simultaneously achieve achromatic and wide-angle imaging.
The intensity distributions of the BAWF metalens in the x-z plane with an incident angle θ of 0° and 80° at five wavelengths of 3800 nm, 3900 nm, 4000 nm, 4100 nm and 4200 nm are shown in Figs. 4(a) and 4(c), respectively. The dashed white line indicates the position of the focal plane. It can be seen that the focal points at different wavelengths and incident angles are located around the same plane, indicating an excellent achromatic and wide-angle imaging capability. Figures 4(b) and 4(d) show the field distributions of the foci along the z-axis. In all cases, the focal positions remain consistent. It proves that the BAWF metalens has the great ability to achieve wide field-of-view and achromatic focusing. In addition, the BAWF metalens exhibits a distinctly large depth of focus, which can be used for nanolithography and optical storage [44].
Fig. 4. Simulation results for the BAWF metalens. (a) Normalized light intensity distributions in the x-z plane at different wavelengths and an incident angle of θ = 0°. (b) Focal field along the z-axis at an incident angle of θ = 0°. (c) Normalized light intensity distributions in the x-z plane at different wavelengths and an incident angle of θ = 80°. (d) Focal field along the z-axis at an incident angle of θ = 80°.
To evaluate the imaging performance of the BAWF metalens, we calculate the full width at half maximum (FWHM) of the focal spot, the focusing efficiency and the focus shift ΔX as a function of the incident angle. The FWHMs of the BAWF metalens are shown in Fig. 5(a), and the dashed black line indicates the theoretical FWHM according to the diffraction limit. The FWHM of the BAWF metalens remains nearly constant as the incident angle θ changes from 0° to 30°, and it is quite close to the diffraction limit. As the incident angle continues to increase, the FWHM gradually increases. At an incident angle of 80°, the FWHM at a wavelength of 3800 nm is not larger than 1.15 λ0, and the focus shape is approximately unchanged (see Supplement 1, Fig. S2), which means that the focusing performance is still acceptable. Figure 5(b) shows the focusing efficiency at different wavelengths as a function of the incident angle. We define the focusing efficiency as the ratio of the optical power in the focal area with a radius of three times the FWHM to the total incident power. We can see that the efficiency remains stable as the incident angle increases. From Fig. 5(b), the focusing efficiency is relatively high at wavelengths of 4100 nm and 4200 nm owing to the higher polarization conversion efficiency of the meta-atoms at these wavelengths. The maximum focusing efficiency is approximately 15% at a wavelength and incident angle of 4200 nm and 20°, respectively. However, the minimum is only 5.2% at a wavelength and incident angle of 3800 nm and 70°, respectively. The low focusing efficiency is due to the inherent spherical aberration introduced by the quadratic phase profile. In addition, the effective focusing aperture of a quadratic phase metalens is smaller than the physical aperture [36]. Beyond the effective focusing aperture, the transmitted light is converted to the evanescent wave and is not focused (see Supplement 1, Note 5). The focusing efficiency can be further improved by using other materials with even lower absorption. Figure 5(c) illustrates the focus shift ΔX in the lateral direction under various conditions. According to Eq. (9), the focus is shifted at an oblique incident angle, and the theoretical shift can be determined by fsinθ. However, the actual focus position slightly deviates from this expectation due to the phase discretization in the metalens design. At wavelengths of 3800 nm and 4200 nm and an incident angle of 40°, the lateral shift ΔX is 3.522 µm and 2.747 µm, respectively, which corresponds to the largest difference in our simulations. The maximum deviation is 0.775 µm, which is smaller than the FWHM of the focus. Moreover, at other incident angles, the difference among the focal lengths at different wavelengths is still smaller than the FWHM. The BAWF metalens is also nearly achromatic in the transverse direction.
Fig. 5. Imaging performance of the BAWF metalens. (a)–(c) FWHM, focusing efficiency and focus shift ΔX under different wavelengths and incident angles.
4. Performance comparison
In this section, we compared the BAWF metalens with other three metalenses, which were a polarization-sensitive CQ metalens, a polarization-insensitive CQ metalens and a DLHA metalens, respectively. The polarization-sensitive CQ metalens was designed by using only the geometric phase, and it was composed of meta-atoms with different rotation angles. The focal length is 28 µm and the diameter is the same as that of the BAWF metalens. Each meta-atom has the same size (P = 1.8 µm, H = 3.5 µm, W = 0.96 µm and L = 0.8 µm). The operating wavelength is 4200 nm. The polarization-insensitive CQ metalens was designed by using cylindrical nanopillars. The quadratic phase profile of the metalens is obtained by the propagation phase, and the focal length is 24.5 µm (see Supplement 1, Note 6). Besides, the DLHA metalens was designed by combining hyperbolic phase profile and propagation phase. The hyperbolic phase profile is mainly achieved by the geometric phase. Again, the lens has the same material and diameter as the BAWF metalens for direct comparison. For the DLHA metalens, the focal length is 24 µm, and the meta-atoms have the same period and height (P = 1.8 µm and H = 3.5 µm). We also performed FDTD simulations to study the focal fields of the DLHA metalens at different wavelengths and incident angles (see Supplement 1, Figs. S7–S11).
Figure 6(a) shows the intensity distributions of the polarization-sensitive CQ metalens in the x-z plane at an incident angle of 30° and wavelengths of 3800 nm, 3900 nm, 4000 nm, 4100 nm and 4200 nm. The dashed line indicates the position of the focal plane. Figure 6(b) shows the distribution profiles of the focus along the z-axis. At an incident angle of 30°, the light field in the x-z plane does not diverge and is still able to focus. These simulations demonstrate the wide field-of-view imaging capability of the metalens with a quadratic phase profile. However, it is clear that the focal positions of the polarization-sensitive CQ metalens do not remain the same as the wavelength changes. In addition, we performed the same simulations of the polarization-insensitive CQ metalens (see Supplement 1, Fig. S13). Again, the results demonstrate that the metalens exhibits significant chromatic aberration. Therefore, it is a challenge for the CQ metalens to realize achromatic imaging due to the absence of phase compensation. The comparison results here emphasize the necessity of the dispersion compensation to realize achromatic focusing.
Fig. 6. Simulations of the polarization-sensitive CQ metalens. (a) Normalized light intensity distributions in the x-z plane at different wavelengths. (b) Focal field along the z-axis. The incident angle is θ=30°.
The normalized focus fields of the BAWF metalens and DLHA metalens at wavelengths of 3800 nm, 4000 nm and 4200 nm and different incident angles are presented in Figs. 7 and 8, respectively. As shown in Fig. 7, the focus of the BAWF metalens remains virtually unchanged when the incident angle θ changes from 0° to 80°. The focal position is shifted, which is consistent with Eq. (9). When θ ≥ 50°, the FWHM of the focus gradually increases. This indicates that the BAWF metalens is capable of imaging over a wide field-of-view. However, it causes a decrease in resolution compared to small field-of-view imaging. For the DLHA metalens, multiple focal spots are witnessed as shown in Fig. 8. When the incident angle increases, this phenomenon becomes more pronounced. This is mainly due to Seidel aberrations, which are closely related to the field-of-view [28]. Therefore, the DLHA metalens exhibits a limited field-of-view and lacks wide-angle imaging capability. Such results also highlight the critical role of employing the quadratic phase profile to achieve the wide field-of-view.
Fig. 7. Light intensity distributions in the focal plane for the BAWF metalens. (a)–(c) Normalized focal fields at different incident angles and wavelengths of 3800 nm, 4000 nm and 4200 nm, respectively.
Fig. 8. Light intensity distributions in the focal plane for the DLHA metalens. (a)–(c) Normalized focal fields at different incident angles and wavelengths of 3800 nm, 4000 nm and 4200 nm, respectively.
5. Conclusions
In summary, we proposed a BAWF single-layer metalens that can realize broadband achromatic imaging over a wide field-of-view by combining quadratic phase profile and propagation phase compensation. First, we explained the formation of the wide field-of-view for the quadratic phase metalens based on analytical ray tracing. Second, we designed the BAWF metalens consisting of rectangular amorphous Si meta-atoms on a SiO2 substrate with an NA of 0.85. We analyzed the imaging performance including the FWHM of the focus, focusing efficiency and focus shift when the incident angle is changed. The BAWF metalens exhibits exceptional wide-angle focusing capability and realizes achromatic focusing in both the optical axis and transverse directions. However, the inherent limitation of the quadratic phase restricts the efficiency, which reaches its maximum of ∼15%. The focusing efficiency can be further improved by using other materials with lower absorption. Finally, we compared the polarization-sensitive and polarization-insensitive CQ metalenses and the DLHA metalens with the BAWF metalens. The CQ metalenses have the ability to achieve wide-angle imaging. However, they are difficult to attain effective achromatic capability. The DLHA metalens exhibits significant off-axis aberrations at large incident angles. In contrast, the BAWF metalens realizes wide field-of-view and achromatic focusing simultaneously owing to the adoption of quadratic phase profile together with propagation phase compensation. However, these improvements are dependent on the increased dispersion range. It is possible to augment the accessible range of compensation phase, such as different meta-atom heights, multiple nanostructures within a meta-atom or combinations of heterogeneous materials. These methods can also be used to enlarge the aperture. As for the metalens processing, it is compatible with conventional nanofabrication techniques, and high-quality fabrication is available [45]. In addition, the metalens design can be applied to other wavebands, and the BAWF metalens has potential applications in unmanned vehicles, infrared detection, integrated optics, etc.
Funding
National Natural Science Foundation of China (52075517).
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.
Supplemental document
See Supplement 1 for supporting content.
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